4RLF 


•MS 


EOMETRY 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


Class 


NUMERICAL  PROBLEMS 

IN 

DESCRIPTIVE  GEOMETRY 

FOR 

CLASS  AND  DRAWING  ROOM 
PRACTISE 

REVISED  EDITION 


By 
G.  M.  BARTLETT,  M.  A. 

Instructor  in  Descriptive  Geometry  and  Drawing  in  the 
University  of  Michigan 


{    UNIVERSITY 

OF 


Published  by  the  Author 

Ann  Arbor,  Michigan 

1907 


Copyright 

G.  M.  BARTLETT 

1905 


The  Ann  Arbor  Press,  Printers 


Cfl 


\  Surfaces  (  Surfaces  of  double  curvatu 

1  Double  /  Surfaces  of  Revolution  I 

)same  p  i  a  n  e;  i 
\  hence,  non-de-  I 
*  velopable. 

\  Curves 
\  line,. 

1  tive  positions  01  rnree  i 
ItfaCeS  /  generatrix  in  I 

1  No  two  consecu-  \ 

—  - 

Bte 

P 
ttP* 

1 

-* 

ViMi 
fci 

-^" 

t 
t 

••  .>, 

2 

p* 

rt 

r 
3' 

-^ 

—  —  ~. 
^ 

r.  P 
3  a 

TI    rt 

5  o- 

/  Generated  by  / 

—  • 

,  — 
o 

1 

•83C 

'E 

m 
4 

2. 

0* 

X) 

p 

cr 

5 

•*  — 
1 

Iaingie  %^/urvcu.  j 
Surfaces  (  Consect 

•^^  ^_ 

D               H3 

•r 

^       g 

1                      CO 

} 

1 

< 

1 

L 

--—  —  ^, 
>      > 

O        O 

Vi         -X 

CLASSIFICATION  OF 

(Right  lines 

Plane  curves  (circle,  ellipse,  parabola,  hyperbola, 
Space  curves  (helix,  spherical  epicycloid,  loxodrc 

re  not  generated 

(A 

•o 
rt 

rt 
•5' 

8 

a 

generated  by  a  rl 
and  making  a  cor 

I 

n 

n 

r. 

r. 
n 
<i 

\i 
i 

( 

( 
( 

< 
( 

n 

i? 

2-» 
U 

O 

1 

o 

p 

3 

a 
o 

ons  intersecting  i 

rt 

1. 
o' 

o 

tions  ol  generatri 
tions  of  generatri 

LINES  AND 

spirals,  roulettes 
ime,  etc.) 

by  a  revolving  line  nor 

0 

«-*> 
•-t 

rt 

0 

•-t 

a 

•o 
P 

cr 

o 
>-•» 

\  Three  curves 
:.  line  touching  a  spac 

istant  angle  with  a  fixe 

\  Two  curves,  one  rl 

]  One  curve,  two  rt.  1 

/  Three  rt.  lines  (hy] 

i  Two  curves  (cylind 

_-^~^— 

0 
a 

rt 
>-t 

a' 
rt 
P 

a 

fV 
0 

a 

[  Two  rt.  lines  (hype 

double  curvature  (heli 

n  common  pt.  May  be 

rt 
n 

P 

a' 

rt 
n 

5 

x  parallel  (cylinders) 
x  intersect  in  common 

SURFACES 

,  etc). 

pi 

rt 

P<  r> 

3 

rt 

o 

o 

P 

crq 

•o 

ro 

6 

P 

? 

5"  c 

«' 

•t. 

D* 

D* 

M 

P 

0 

3 

^  ( 

rt 

o 

P 

0 

rt 

o 

rt 

P 

3 

O 

o 

0    rt 

c* 

p 

a 

5' 

'ff  P 
2.  B 

O 
>n 

2- 
0 

0 

^ 

£ 

rt 

- 

rt 

M 

P 

o 

3 
rt 

O 

0 

rt 

o* 

0 

0    P 

p.. 

rt 

rt 

a 

t? 

B> 

3* 

P 

a 
P 

(A 

»< 

o 

ABBREVIATIONS 

alt.  =  altitude. 

bet.  =  between. 

const.  =  construct. 

cyl.  =  cylinder. 

dia.  =  diameter. 

dist.  =  distance  or  distant. 

proj.  =  projection. 

pt.  =  point. 

rt.  =  right. 

tang.  =  tangent. 

G.  L.  =  ground  line. 

H  ==  horizontal  or  horizontal  plane, 

P  =  profile  or  profile  plane. 

V  =  vertical  or  vertical  plane. 

1  =  perpendicular. 

||  =  parallel. 

L  =  angle. 

/ 8  —  angles. 


ELEMENTARY  PROPOSITIONS 

1.  The  distance  of  a  point  from  the  H  plane  is  the 
distance  of  its  V  projection   from  the  ground  line.     The 
distance  of  a  point  from  the  V  plane  is  the  distance  of  its 
H  projection  from  the  ground  line. 

2.  If  any  two  projections  of  a  point  are  given,  the 
point  is  fully  determined,  and  the  third  projection  may  be 
found  from  the  above  principle. 

3.  If  a  line  is  parallel  to  either  plane  of  projection,  its 
projection  upon  the  other  plane  of  projection  is  parallel 
to  the  ground  line. 

4.  If  a  line  is  parallel  to  either  plane  of  projection,  it 
will  be  projected  upon  that  plane  in  its  true  length. 

5.  If  two   intersecting   lines   are   each   parallel   to  the 
same  plane  of  projection,  the  angle  between  the  lines  will 
be  projected  upon  that  plane  in  its  true  magnitude. 

6.  If  a  point  lies  in  a  given  line,  its  projections  will 
lie  in  the  corresponding  projections  of  the  line. 

7.  If  two  lines  are  parallel  in  space,  their  correspond- 
ing projections  are  parallel. 

8.  The  H  and  V  traces  of  a  plane  always  intersect  the 
ground  line  at  the  same  point. 

9.  If  any  two  traces  of  a  plane  are  given,  the  plane 
is  fully  determined. 

10.  If  a  plane  is  parallel  to  the  ground  line,  its  traces 
arc  parallel  to  the  ground  line,  and  conversely. 

n.  If  a  plane  is  perpendicular  to  either  plane  of  pro- 
jection, its  trace  upon  the  other  plane  is  perpendicular  to 
the  ground  line,  and  conversely. 

12.  If  a  line  lies  in  a  given  plane,  its  H  piercing  point 
lies  in  the  H  trace  of  the  plane,  and  its  V  piercing  point 
lies  in  the  V  trace  of  the  plane. 


6  PROBLEMS    IN 

13.  If  a  line  is  perpendicular  to  a  plane,  its  projections 
will   be   perpendicular  to   the   corresponding  traces  of  the 
plane,  and  conversely. 

14.  If  a  line  lies  in  a  plane  and  is  parallel  to  H  (or  V),. 
its  H  (V)  projection  is  parallel  to  the  H  (V)  trace  of  the 
plane,  and  its  V   (H)   projection  is  parallel  to  the  ground 
line. 

15.  If  two  planes  are  parallel  in  space,  their  corres- 
ponding traces  will  be  parallel,  and  conversely. 

1 6.  If  a  point  in  space  be  rotated  about  a  line  either 
lying  in  or  parallel  to  H,  the  H  projection  of  the  point  will 
always  lie  in  the  same  perpendicular  to  the  H  projection 
of  the  line. 

17.  If  a  point  in  space  be  revolved  into  H  about  a  line 
lying  in  H,  its  position  in  H  will  be  at  a  distance  from  the 
axis  equal  to  the  hypotenuse  of  a  right  triangle  whose  legs 
are  respectively  the'  distance  from  the  H  projection  of  the 
point  to  the  H  projection  of  the  line,  and  the  distance  from 
the  V  projection  of  the  point  to  the  ground  line. 

1 8.  If  a  line  not  parallel  to  H  be  revolved  about  an 
axis  intersecting  it  and  parallel  to  H,  into  a  position  where 
both  are  parallel  to  H,  any  point  in  the  revolved  line  will 
be  horizontally  projected   at  a  distance   from  the   H  pro- 
jection of  the  axis  equal  to  the  hypotenuse  of  a  right  tri- 
angle whose  base  is  the  distance  of  the  H  projection  of  the 
point    (before   revolution)    from   the   H   projection   of  the 
axis,  and  whose  altitude  is  the  distance  of  the  V  projection 
of  the  point  (before  revolution)   from  the  V  projection  of 
the  axis. 


PROPOSITIONS  RELATING  TO  WARPED  SURFACES 

1.  The  rectilinear  elements  of  an  hyperbolic  paraboloid 
divide  the  directrices  proportionally,  and  conversely. 

2.  If  two  right  lines  be  divided  into  any  number  of 
proportional  parts,  the  right  lines  joining  the  correspond- 


DESCRIPTIVE  GEOMETRY  7 

ing  points  of  division  will  line  in  a  system  of  parallel  planes, 
and  hence  be  elements  of  an  hyperbolic  paraboloid,  the 
plane  directer  of  which  is  parallel  to  any  two  of  these  lines. 

3.  If   any   two   rectilinear   elements   of   an   hyperbolic 
paraboloid   be   taken  as   directrices,   with   a   plane   directer 
parallel  to  the  first  directrices,  and  a  surface  be  thus  gen- 
erated, it  will  be  identical  with  the  first  surface. 

4.  Thru   any   point   of   an   hyperbolic   paraboloid,   two 
rectilinear  elements  can  always  be  drawrn. 

5.  If  two  warped  surfaces  having  two  directrices  have 
a  common  plane  directer,  a  common  rectilinear  element  and 
two  common  tangent  planes,  the  points  of  contact  being  on 
the  common  element,   they  will  be  tangent  all  along  this 
element. 

6.  If  two  warped  surfaces  have  an  element  in  common 
and  are  tangent  to  each  other  at  three  points  of  the  same, 
they  are  tangent  along  the  entire  element. 


REPRESENTATION  OF  POINTS,  LINES  AND  PLANES 

i.     Show   the  projs.   of   the   following  points   properly 
lettered  and  with  distances  given. 

The  pt.  A,  i"  behind  V,  i^"  below  H. 

The  pt.  B,  lying  in  V,  i"  below  H. 

The  pt.  C,  3"  in  front  of  V,  i"  above  H. 

The  pt.  D,  i"  behind  V,  lying  in  H. 

The  pt.  E,  2"  behind  V,   iy2"  below  H. 

The  pt.  F,  i"  in  front  of  V,  i"  below  H. 

The  pt.  G,  lying  in  V,  2"  above  H.' 

The  pt.  J,  i"  in  front  of  V,  lying  in  H. 

The  pt.  K,  lying  in  V,  lying  in  H. 

The  pt.  L,  in  3rd  quadrant,  i"  from  II,  2"  from  V. 

The  pt.  M,  in  2nd  quadrant,  3"  from  H,  2"  from  V. 

The  pt.  N,  in  ist  quadrant,  il/2"  from  H,  3^"  from  V. 

The  pt.  P,  in  4th  quadrant,  4"  from  H,  i"  from  V. 


8  PROBLEMS    IN 

2.  State  in  what  quadrant  each  of  the  points  shown  in 
the  figure  are  located,  and  whether  the  point  is  nearer  H 
or  V. 


V*     »  !        J      r 

/^  /  ^^ 

*i     xl     x 

?ri 

f. 

*^i 

e 

"      ^V       i 

x|              x^                     * 

^;                                       Sj 

L 

\\ 

»     ^ 

d' 


3.  Show  the  projs.  of  the  lines 
AB,  ||  to  H,  ||  to  V,  in  3rd  quadrant. 

CD,  ||  to  H,  perpendicular  to  V,  in  2nd  qaudrant. 
EF,  ||  to  H,  inclined  to  V,  in  ist  quadrant. 
GH,  inclined  to  H,  ||  to  V,  in  ist  quadrant. 
JK,  1  to  H,  parallel  to  V,  in  2nd  quadrant. 
MN,  inclined  to  H,  inclined  to  V,  in  ist  quadrant. 
OP,  inclined  to  both  planes  of  proj.  and  in  a  plane  per- 
pendicular to  the  G.  L.,  in  ist  quadrant. 

QR,  inclined  to  V,  and  lying  in  H.  beyond  G.  L. 
ST,  inclined  to  H,  and  lying  in  V  above  G.  L. 
UV,  lying  in  both  H  and  V. 

4.  Const,  the  projs.  of  two  lines,  AB  and  AC,  inter- 
secting in  A,  one  ||  to  H,  the  other  ||  to  V. 

5.  Show  the  projs.  of  a  line  joining  a  point  A  in  the 
2nd  quadrant  with  a  point  B  in  the  3rd  quadrant. 

6.  Show  the  projs.  of  a  line  joining  a  point  C  in  the 
4th  quadrant  with  a  point  D  in  the  ist  quadrant. 


DESCRIPTIVE   GEOMETRY 


7.  Describe  the  situation  of  the  following  lines  with 
respect  to  the  ground  line,  the  planes  of  projection  and  the 
quadrants. 


m  — 


~nj 


P' 


s. 


~n'  v 


4 


f 


8.  If  the  H  proj.  of  a  line  is  ||  to  the  G.  L.,  what  con- 
clusion do  you  draw  (a)  as  to  the  position  of  the  line?  (b) 
as  to  its  intersection  with  the  V  plane?   (c}   as  to  the  V 
trace  of  any  plane  passed  through  the  line? 

9.  If  a  right  line  lies  in  a  given  plane,  what  conclusion 
do  you  draw  as  to  the  points  where  the  line  pierces  the  H 
and  V  planes  respectively? 

10.  Represent  by  its  traces  the  planes: 

aAa',  1  to  both  H  and  V. 

bBb',  inclined  to  H ;  1  to  V. 

cCc',  1  to  H;  inclined  to  V. 

dDd',  inclined  to  both  H  and  V. 

eEe'..  1  to  H;  ||  to  V. 

fFf,  ||  to  H;  1  to  V. 

gGg',  ||  to  G.  L-,  but  not  passing  thru  it. 

hHh',  containing  the  G.  L. 

IT.  Describe  the  situation  of  the  following  planes  with 
respect  to  the  ground  line,  the  planes  of  projection,  and  the 
quadrants. 


[O 


PROBLEMS    IN 


T 


Fig.  5. 


R- 
R- 


-r 

k 

-r 


12.  If  the  H  trace  of  a  plane  is  _L  to  the  G.  L.,  what 
conclusion  do  you  draw  (a) as  to  the  position  of  the  plane? 
(fr)  as  to  the  Z  between  the  plane  and  the  H  plane?  (c) 
as  to  the  V  proj.  of  any  line  lying  in  the  plane?  (d)  as  to 
the  /  bet.  the  plane  and  the  V  plane? 


PROBLEMS   RELATING   TO   THE    POINT,  LINE  AND 

PLANE 

13.  Find  the  H  and  V  piercing  pts.  of  the  line  MD, 
Fig.  4. 

14.  Find  the  H  and  V  piercing  pts.  of  the  line  SB, 
Fig.  4.  Const,  double  size. 

15.  Find  the  H  and  V  piercing  pts.  of  the  line  DN, 
Fig.  5.  Const,  double  size. 

1 6.  Find  the  H  and  V  piercing  pts.  of  the  line  BM, 
Fig.  6. 

17.  Find  the  H  and  V  piercing  pts.  of  the  line  CD, 
Fig.  4. 

18.  Find  the  H  and  V  piercing  pts.  of  the  line  BM. 
Fig.  4. 


DESCRIPTIVE   GEOMETRY 


I  I 


Fig.  4. 


A': 


s*e 


r— TT 


12 


PROBLEMS    IN 


INSCRIPTIVE   GEOMETRY 


J4  PROBLEMS    IN 

19.  Find  the  H  and  V  piercing  pts.  of  the  line  MD, 
Fig.  6.     Const,  double  size. 

20.  Find  the  H  and  V  piercing  pts.  of  the  line  AC, 
Fig.  6. 

21.  Const,  the  projs.  of  the  line  that  pierces  H  at  b 
and  V  at  m',  Fig.  6. 

22.  Const,  the  projs.  of  the  line  that  pierces  H  at  w 
and  V  at  d',  Fig.  6. 

23.  Const,  the  projs.  of  the  line  that  pierces  H  at  c 
and  V  at  m',  Fig.  4. 

24.  Const,  the  projs.  of  the  line  that  pierces  H  at  a 
and  V  at  w',  Fig.  4. 

25.  Find  the  length  of  the  line  EC,  Fig.  4. 

26.  Find  the  length  of  SB,  Fig.  4. 

27.  Find  the  length  of  MD,  Fig.  4,  by  two  methods. 

28.  Find  the  length  of  EN,  Fig.  6,  by  two  methods. 

29.  Find  the  length  of  CM,  Fig.  6,  by  two  methods. 

30.  Find  the  length  of  DE,  Fig.  6,  by  two  methods. 
Const,  double  size. 

31.  Find  a  point  N  on  the  line  EC,  Fig.  4,  that  is  2" 
from  E.    Give  the  dist.  of  n  from  e. 

32.  Find  a  point  O  on  the  line  MD,  Fig.  4,  2"  from 
M.    Give  the  dist.  of  o  from  m. 

33.  Find  a  point  Q  on  the  line  EN,  Fig.  6,  i"  from  N. 
Give  the  dist.  of  q  from  n. 

34.  Find  the  Z   which  MD,  Fig.  4  ,makes  with'H  and 
with  V. 

35.  Find  the  Z   which  EC,  Fig.  4,  makes  with  H,  and 
with  V. 

36.  Find  the  Z  which  EN,  Fig.  6,  makes  with  H,  and 
with  V. 

37.  The  H  proj.  of  a  line  is  dn,  Fig.  5.     Its  H  trace 
is  at  d.    The  line  makes  an   Z   of  30°  with  H.     Const,  its 
V  proj.    Give  dist.  of  n'  from  G.  L. 

38.  The  H  proj.  of  a  line  is  ce,  Fig.  6.     Its  H  trace 


DESCRIPTIVE   GEOMETRY  15 


is  at  c.     The  line  makes  an    Z   of  22j40  with  V.     Const. 
its  V  proj.    Give  dist.  of  c'  from  G.  L. 

39.  Find  the  point  A  in  the  line  DN,  Fig.  5,  equally 
dist.  from  H  and  V.    Give  dist.  da.    Const,  double  size. 

40.  Same  as  above,  but  equally  dist.  from  H  and  P. 
Take  the  P  plane  thru  N.    Give  dist.  da. 

41.  Pass  a  plane,  tTt',  thru  A,  M  and  D,  Fig.  4.    Give 
Z   bet.  traces  on  drawing. 

42.  Pass  a  plane,  uUu',  thru  B,  C  and  E,  Fig.  6.    Give 
Z   bet.  traces  on  drawing. 

43.  Pass  a  plane,  uUu',  thru  B,  D  and  M,  Fig.  6.    Give 
Z   bet.  traces  on  drawing. 

44.  Pass  a  plane,  sSs',  thru  the  point  C  and  the  line 
AM,  Fig.  4.     Give  Z  bet.  traces  on  drawing. 

45.  Pass  a  plane,  tTt',  thru  the  point  M  and  the  line 
AB,  Fig.  4.    Give  Z  bet.  traces  on  drawing. 

46.  Pass  a  plane,   sSs',   thru   the  lines  EC  and   MC, 
Fig.  4.     Give  Z   bet.  traces  on  drawing. 

47.  Pass  a  plane,  uUu',  thru  the  'lines  BM  and  BE, 
Fig.  6.    Give  Z  bet.  traces  on  drawing. 

48.  Pass   a  plane,   sSs',   thru   the   lines   BC   and   DC, 
Fig.  4.    Give  Z  bet.  traces  on  drawing. 

49.  Pass  a  plane,  kKk',  thru  the  lines  CM  and  DM, 
Fig.  6.     Give    Z    bet.  traces  on  drawing. 

50.  Pass  a  plane,  tTt',  thru  MD  and  a  line  ||  to  MD 
thru  C,  Fig.  4.    Give  Z  bet.  traces  on  drawing. 

51.  Pass  a  plane,  kKk',  thru  EN'  and  a  line  ||  to  EN 
thru  B,  Fig.  6.    Give  Z  bet.  traces  on  drawing. 

52.  Pass  a  plane,  sSs',  thru  the  line  AB  and  a  line  || 
to  the  G.  L.   thru   M,  Fig.  4.     Give  dist.  bet.  traces  on 
drawing. 

53.  Pass  a  plane,  kKk',  thru  the  line  AB  and  a  line  || 
to  the  G.  L.  thru  E,  Fig.  6.    Give  dist.  bet.  traces  on  draw- 
ing. 

54      Find  the  value  of  the  Z   EMD,  Fig.  4. 
55.     Find  the  value  of  the   Z    SDM,  Fig.  4. 


1 6  PROBLEMS    IN 

56.  Find  the  value  of  the    Z    BMN,   Fig.  6.     Draw 
double  size. 

57.  Find   the  value  of  the    Z    BEN,   Fig.   6.     Draw 
double  size. 

58.  Find   the   value   of  the    Z    ABS,   Fig.   4.      Draw 
double  size. 

59.  Find  the  value  of  the    Z    ABM,   Fig.   6.     Draw 
double  size. 

60.  Find  the  value  of  the   Z   ADN,  Fig.  6. 

61.  Find  the   Z   bet.  the  lines  SE  and  SB,  Fig.  4. 

62.  Find   the    Z    bet.    ON   and   OM,    Fig.   8.     Draw 
double  size. 

63.  Find  the    Z    in  space  bet.  the  traces  of  the  plane 
rRr',  Fig.  4. 

64.  Find  the   Z   in  space  bet.  Uu  and  Uu',  Fig.  5. 

65.  Find  the    Z    in  space  bet.  the  traces  of  the  plane 
gGg'..  Fig.  4. 

66.  Show  the  true  size  and  shape  of  the  triangle  BCD, 
Fig.  4. 

67.  Show  the  true  size  and  shape  of  the  triangle  BMN. 
Fig.  6.    Draw  double  size. 

68.  Find  the  true  size  and  shape  of  the  triangle  OGD, 
Fig.  8.    Give  lengths  of  the  three  sides. 

69.  Find  the  true  size  and  shape  of  the  triangle  ADG, 
Fig.  8. 

70.  Find  the  bisector,   CN,   of  the    Z    MCD,   Fig  6. 
Give   Z   m'c'n'. 

71.  Find  the  bisector,  DN,  of   Z    SDM,  Fig.  4.     Give 
Z   sfd'n'. 

72.  Find  the  bisector,  EF  of    Z    AEN,  Fig.  6.     Give 
Z   a'e'f.     Draw  double  size. 

73.  Find  the  bisector,  EF,  of   Z    BEN,  Fig.  6.     Give 
Z   b'e'f.    Const,  double  size. 

74.  Find  the  bisector,  BN,  of   Z    ABS,  Fig.  4.     Give 
Z  tfVri.    Const,  double  size. 


DESCRIPTIVE  GEOMETRY  ^17 

75.  Find  the  bisector,  BN,  of  Z   ABM,  Fig.  6.     Give 
Z   a'b'n'.    Const,  double  size. 

76.  Find  the  bisector,  SN,  of   Z    ESB,  Fig.  4.     Give 
Z  esn. 

77.  Find  the  bisector,  EN,  of   Z    AEC,  Fig.  4.     Give 
Z  (tefri. 

78.  Thru  B,  Fig.  4,  pass  a  line  BN,  making  an   Z   of 
45°  with  EC.    Give  Z  £wr.     (Two  solutions.) 

79.  Thru  A.  Fig.  4,  pass  a  line,  AN,  1  to  SB.     Give 
Z   anb. 

80.  Find  the  pt.  Q.  where  the  line  MD  pierces  the  plane 
rRr',  Fig.  4.     Give  dist.  bet.  q  and  q'  on  drawing.     Const, 
double  size. 

81.  Find  the  pt.  K  where  the  line  AB,  Fig.  8,  pierces 
the  plane  rRr'.     Give  dist.  bet.  k  and  k'  on  drawing. 

82.  Find  the  pt.  Q  where  the  line  SM  pierces  the  plane 
kKk',  Fig.  4.    Give  dist.  bet.  q  and  q'  on  drawing.     Const, 
double  size. 

83.  Find  the  pt.  F  where  the  line  AB  pierces  the  plane 
rRr',  Fig.  4.      Give  dist.  BF. 

84.  Find  the  pt.  F  where  a  line  thru  D  ||  to  the  G.  L. 
pierces  the  plane  uUu',  Fig.  5.     Give  dist.  DF. 

85.  Find  the  pt.  F,  where  the  line  AB  pierces  the  plane 
kKk',  Fig.  4.     Give  dist.  AF.     Draw  double  size. 

86.  Find  the  pt.  F,  where  the  line  DN  pierces  the  plane 
sSs',  Fig.  5.    Give  dist.  fd. 

87.  Find  the  pt.  F,  where  the  line  DN  pierces  a  plane 
1  to  the  ground  line  and  passing  thru  W,  Fig.  5. 

88.  Find  the  pt.  F,  where  the  line  CD  pierces  the  plane 
kKk',  Fig.  4.     Give  dist.  fd.    Draw  double  size. 

89.  Find  the  pt.  F,  where  the  line  AC  pierces  the  plane 
rRr',  Fig.  8.    Give  dist.  /'a'. 

90.  Find  the  pt.  F,  where  the  line  BM  pierces  the  plane 
rRr',  Fig.  4.     Give  dist.  fb.    Draw  double  size. 

91.  Find  the  pt.  F,  where  EC  pierces  the  plane  of  MD 


1 8  PROBLEMS    IN 

• 

and  BD,  Fig.  4,  without  constructing  the  traces  of  the  plane. 
Give  dist.  cf.     Draw  double  size. 

92.  Find  the  pt.  F,  where  BE  pierces  the  plane  of  MN 
and  CN,  Fig.  6,  without  constructing  the  traces  of  the  plane. 
Give  dist.  bf. 

93.  The  pt.  Q  lies  in  the  plane  rRr',  Fig.  5.     Its  V 
proj.,  q' ',  is  given.    Find  q,  and  give  its  dist.  from  the  G.  L. 

94.  The  V  proj.  of  a  pt.  Q  lying  in  the  plane  tTt'  is 
given  at  q',  Fig.   5.     Find  q,  and  give  its  dist.   from  the 
ground  line.     Draw  double  size. 

95.  The  line  QO  lies  in  the  plane  rRr',  Fig.   5.     Its 
V  proj.,  q'o',  is  given.     Locate  qo. 

96.  If  the  line  QO  lies  in  the  plane  sSs'  and  q'o'  is 
given,  Fig.  5,  locate  qo. 

*97.  If  ab  is  the  H  proj.  of  a  line  lying  in  the  plane 
rRr',  Fig.  4,  locate  a'V. 

^98.  If  Gm  is  the  H  proj.  of-  a  line  lying  in  rRr',  Fig. 
4,  locate  g'm'.  (The  pt.  G  lies  in  the  ground  line.) 

*99.  If  bed  is  the  H  proj.  of  a  triangle  BCD  lying  in 
the  plane  sSs',  Fig.  6,  find  its  V  proj.  and  the  true  size 
and  shape  of  the  triangle. 

TOO.     Thru  D,  pass  a  line,  MS,    1    to    ullu',    Fig.  5. 
Const,  double  size. 

101.  Thru   CD,   pass   a   plane,  tTt',  1  to  rRr',  Fig.  4. 
Give  the  dist.  RT. 

102.  Thru  BM,  Fig.  6,  pass  a  plane,  uUu',  1  to  sSs'. 
Give  dist.  US. 

103.  Thru  EM,  Fig.  4,  pass  a  plane  sSs',  1  to  kKk'. 
Give  dist.  SK. 

104.  Find  the  dist.  from  the  pt.  D  to  the  plane  uUu', 

Fig-  5- 

105.  Find  the  dist.  from  the  pt.  E  to  the  plane  sSs', 

Fig.  6. 


*In  solving  Probs.  97,  98  and  99,  no  attention  should  be  paid 
to  the  V  proj's.  as  given  in  the  figure. 


DESCRIPTIVE  GEOMETRY  19 

106.  Find  the  dist.  from  the  pt.  D  to  the  plane  uUu' 
Fig.  7. 

107.  Find  the  dist.  from  the  pt.  D  to  the  plane  tTt', 
Fig.  5.    Draw  double  size. 

108.  Find  the  dist.  from  the  pt.  N  to  the  plane  sSs', 
Kg.  5- 

109.  Find  the  dist.  from  the  pt.  D  to  the  plane  tTt', 
Fig.  7. 

no.  Find  the  dist.  from  the  pt.  N  to  a  plane  thru  W, 
_L  to  the  G.  L.,  Fig.  5. 

in.  Find  the  clist.  from  one  corner  of  a  3"  cube  to 
the  plane  of  the  three  adjacent  corners. 

112.  Find  the  clist.  from  one  corner  of  a  2,y2"  cube  to 
the  plane  of  the  three  adjacent  corners. 

113.  Project  the  line  DN  upon  the  plane  uUu',  Fig.  5, 
and  show  the  true  position  of  the  proj.  D^^  with  respect 
to  the  H  trace. 

114.  Project  the  line  BE,  Fig.  6,  upon  the  plane  rRr', 
and  show  the  true  relation  of  this  proj.  B1E1,  to  the  H  trace. 

1-15.  Project  the  line  DN,  Fig.  5,  upon  the  plane  sSs', 
and  show  the  true  relation  of  this  proj.  DjN^  to  the  H  trace. 

116.  Project  the  line  DN,  Fig.   5,  upon  a  plane  thru 
W,  1  to  the  G.  L.,  and  show  the  true  relation  of  this  proj. 
DjNj,  to  the  V  trace.     Draw  double  size. 

117.  Project  the  triangle  BMN,  Fig.  6,  upon  the  plane 
sSs',  and  show  the  true  form  of  this  proj.  B1M1N1,  giving 

/  M^B!. 

118.  Project  the  triangle  ABG,  Fig.  8,  upon  the  plane 
sSs',  and  show  the  true  form  of  this  proj.  A1B1G1,  giving 

z  GAB,. 

,119.  Thru  K,  Fig.  4,  const,  the  traces  of  a  plane  ||  to 
rRr'.  Find  the  dist.  bet.  these  two  planes.  Const,  double 
size. 

1 20.  Thru  K,  Fig.  4,  const,  the  traces  of  a  plane  ||  to 
gGg'.  Find  the  dist.  bet.  the  two  planes. 


20  PROBLEMS   IN 

121.  Thru  C,  Fig.  4,  pass  a  plane  tTt'  1  to  the  line  MD. 
Give  dist.  KT. 

122.  Thru  M,  Fig.  4,  pass  a  plane,  tTt',  1  to  the  line 
BS.     Give  dist.  KT. 

123.  Thru  D,  Fig.  8,  pass  a  plane,  uUu',  1  to  the  line 
GO.     Give  dist.  SU. 

124.  Thru  C,  Fig.  4,  pass  a  plane  tTt',  1  to  the  line 
AS.     Give  dist.  bet.  H  and  V  traces  on  drawing. 

125.  Thru  G,  Fig.  8,  pass  a  plane  uUu',  1  to  the  line 
AC.     Give  dist.  bet.  H  and  V  traces  on  drawing. 

126.  Thru  D,  Fig.  4,  pass  a  plane  tTt',  1  to  the  line 
BC.     Give  dist.  RT. 

127.  Const,  the  proj.s  of  a  circle  whose  center  lies  in 
the  line  MD,  Fig.  4,  whose  plane  is  1  to  MD,  and  whose 
circumference  passes  thru  B.     Give  major  and  minor  axes 
of  H  proj. 

128.  Thru  D,  Fig.  4,  pass  a  plane,  tTt',  ||  to  the  lines 
BC  and  EM.     Give  dist.  GT. 

129.  Thru  N,  Fig.  6,  pass  a  plane  uUu',  ||  to  the  lines 
BM  and  CE.     Give  dist.  TU. 

130.  Thru  O,  Fig.  8,  pass  a  plane  uUu',  ||  to  the  lines 
GM  and  DE.     Give  dist.  SU. 

131.  Thru  D,  Fig.  4,  pass  a  plane  tTt',  ||  to  the  lines 
AB  and  SM.     Give  dist.  bet.  H  and  V  traces  on  drawing. 

132.  Thru  D,  Fig.  6,  pass  a  plane  tTt',  ||  to  the  lines 
AB  and  CE.     Give  dist.  bet.  H  and  V  traces  on  drawing 

133.  Thru  B,  Fig.  4,  pass  a  plane  sSs',  ||  to  the  lines 
CD  and  SM.     Give  dist.  KS. 

134.  Thru  B,  Fig.  6,  pass  a  plane  uUu',  ||  to  the  lines 
MD  and  AE.     Give  dist.  SU. 

135.  Thru  D,   Fig.  4,  pass  a  plane,  tTt',   ||   to  rRr'. 
Give  dist.  RT. 

136.  Thru  N,  Fig.  5,  pass  a  plane,  kK!  ',  ||  to  uUu'. 
Give  dist.  UK. 

137.  Thru  M,  Fig.  4,  pass  a  plane,  sSs',   ||  to  kKk'. 
Give  dist.  SK. 


DESCRIPTIVE:  GEOMETRY  21 

138.  Pass  a  plane,  sSs',   ||  to  rRr',  Fig.  4,  and   ij£" 
from  it.     Give  dist.  SR. 

139.  Pass  a  plane,  kKk',  ||  to  uUu',  Fig.  5,  and  ij^" 
from  it.     Give  dist.  KU. 

140.  Pass  a  plane,  tTt',  ||  to  kKk',  Fig.  4,  and  i"  from 
it.     Give  dist.  TK. 

141.  Pass  a  plane,  tTt',  thru  the  line  SB,   ||  to  MD, 
Fig.  4.    Give  dist.  KT. 

142.  Pass  a  plane,  kKk',  thru  the  line  CB,  ||  to  MN, 
Fig.  6.     Give  dist.  KS. 

143.  Pass  a  plane,  tTt',  thru  the  line  MD,  ||  to  AB, 
Fig.  4.     Give  dist.  bet.  traces  on  drawing. 

144.  Pass  a  plane,  sSs',  thru  the  line  AB,   ||  to  SM, 
Fig.  4.     Give  dist.  bet.  traces  on  drawing.     Draw  double 
size. 

145.  Pass  a  plane,  tTt',  thru  the  line  AS,   ||  to  MC, 
Fig.  4.    Give  dist.  KT. 

146.  Pass  a  plane,  uUu',  thru  the  line  MC,  ||  to  AS, 
Fig.  4.    Give  dist.  KU. 

147.  Find  the  shortest  dist.  from  the  pt.  C  to  the  line 
MD,  Fig.  4. 

148.  Find  the  shortest  dist.  from  the  pt.  M  to  the  line 
DN,  Fig.  6. 

149.  Find  the  shortest  dist.  from  the  pt.  D  to  the  line 
AB,  Fig.  8. 

150.  Find  the  shortest  dist.  from  the  pt.  C  to  the  line 
BD,  Fig.  4. 

151.  Find  the  shortest  dist.  from  the  pt.  M  to  the  line 
AS,  Fig.  4. 

152.  Find  the  shortest  dist.  from  the  pt.  S  to  the  line 
AB,  Fig.  4. 

153.  Find  the  shortest  dist.  from  the  pt.  B  to  the  line 
SE,  Fig.  4. 

154.  Find  the  dist.  bet.  the  ||  lines  DN  and  EO,  Fig.  6. 

155.  Find  the    Z    which  the  line  DN,  Fig.  5,  makes 
with  the  plane  uUu'. 


22  PROBLEMS    IN 

156.  Find  the    Z    which  the  line   DG,   Fig.   8,   makes 
with  the  plane  sSs'. 

157.  Find  the    Z    which  the  line  DN,   Fig.   5,  makes 
with  the  plane  sSs'.     Draw  double  size. 

158.  Find  the    Z    which  the  line   DN,   Fig.   5,  makes 
with  a  plane  thru  W,  1  to  the  G.  L. 

159.  Find  the    Z    which  the  line  AB,   Fig.  4,   makes 
with  the  plane  rRr'. 

1 60.  Find  the    Z    which  the  line  AS,   Fig.  4,   makes 
with  the  plane  rRr'. 

161.  Find  the  intersection  of  the  planes  gGg'  and  rRr'; 
Fig.  4. 

162.  Find  the  intersection  of  the  planes  rRr'  and  sSs', 
Fig.  6. 

163.  Find  the  intersection  of  the  planes  gGg'  and  kKk'r 
Fig.  4. 

164.  Find  the  intersection  of  the  planes  rRr'  and  tTt'r 
-Fig.  6. 

165.  Find  the  intersection  of  the  planes  sSs'  and  tTt'r 
Fig.  6.  Draw  double  size. 

166.  Find  the  intersection  of  the  planes  tTt'  and  sSs'. 
Fig.  5.  Give  dist.  bet,  proj.s  on  drawing. 

167.  Find  the  intersection  of  the  planes  sSs'  and  tTt'- 
Fig.  7.  Give  dist.  bet.  proj.s  on  drawing. 

168.  Find  the  intersection  of  the  planes  uUu'  and  rRr'. 

Fig.  5- 

169.  Find  the  intersection  of  the  planes  uUu'  and  rRr' 

Fig.  7- 

170.  Find  the  intersection  of  the  plane  uUu',  Fig.  5, 
with  a  plane  ||  to  H  whose  V  trace  is  Tt'. 

171.  Find  the  intersection  of  uUu',  Fig.  5,  with  a  plane 
containing  the  G.   L.,  making  an    Z    of  30°   with  H,  and 
passing  thru  the  ist  and  3rd  quadrants. 

172.  Pass  a  line  thru  M,  Fig.  4,  and  ||  to  both  gGg' 
and  rRr'. 


DESCRIPTIVE    GEOMETRY  23 

173.  Pass  a  line  thru  D,  Fig.  8,  and   ||   to  both  sSs' 
and  tTt'. 

174.  Find  the  Z  bet.  the  planes  gGg'  and  rRr',  Fig.  4. 

175.  Find  the   Z   bet.  the  planes  rRr'  and  sSs',  Fig.  6. 

176.  Find  the   Z   bet.  the  planes  sSs'  and  tTt',  Fig.  8. 

177.  Find  the   Z   bet.  the  planes  sSs'  and  tTt',  Fig.  5. 

178.  Find  the   Z   bet.  the  planes  sSs'  and  tTt',  Fig.  7. 

179.  Find  the   Z   bet.  the  planes  sSs'  and  rRr',  Fig.  5. 

180.  Find  the   Z   bet.  the  planes  tTt'  and  uUu',  Fig.  5. 

181.  Find  the   Z   bet.  the  plane  rRr',  Fig.  4,  and  each 
plane  of  proj. 

182.  Find  the  Z  bet.  the  plane  uUu',  Fig.  5,  and  each 
plane  of  proj. 

183.  Find  the   Z   bet.  the  plane  kKk',  Fig.  4,  and  each 
plane  of  proj. 

184.  If  fcr/ Fig.  4,  is  the  V  trace  of  a  plane,  tTt',  that 
makes  an  Z  of  75°  with  V,  const,  the  H  trace  of  the  plane. 
(Two  solutions.)     Give  the   Z  bet.  H  trace  and  G.  L. 

185.  If  iv n,  Fig.  6,  is  the  H  trace  of  a  plane,  uUu',  that 
makes  an   Z   of  60°  with  H,  what   Z   does  the  plane  make 
with  V? 

1 86.  If  sc,  Fig.  4,  is  the  H  trace  of  a  plane,  tTt',  that 
makes  an    Z   of  45°  with  H,  const,  the  V  trace,  giving  its 
Z    with  G.  L.     (Two  solutions.) 

187.  If  d'm',  Fig.  8,  is  the  V  trace  of  a  plane,  kKk', 
that  makes  an    Z    of  45°  with  V,  what    Z    does  the  plane 
make  with  H? 

188.  If  the  V  trace  of  a  plane,  sSs',  is  1  to  the  G.  L., 
and  the  plane  makes  an    Z    of  60°   with  V,  const,  the  H 
trace.      (Two  solutions.) 

189.  If  ab,  Fig.  4,  is  the  H  trace  of  a  plane,  uUu',  that 
makes  an  Z  of  60°  with  H,  and  traverses  the  3rd  quadrant, 
const,  the  V  trace,  giving  its  dist.  from  the  G.  L. 

190.  If  we,  Fig.  6,  is  the  H  trace  of  a  plane  kKk'  that 
traverses  the  4th  quadrant  and  makes  an   Z   of  525/2°  with 
H,  const,  the  V  trace,  giving  its  dist.  from  the  G.  L. 


24  PROBLEMS    IN 

191.  Find  the  pt.  F,  in  H,  equally  dist.  from  D,  W  and 
O,  Fig.  8. 

192.  Find  a  pt.  W  in  V,  equally  dist.  from  S,  M  and 
D,  Fig.  4 

*IQ3.  Take  dn  as  the  H  proj.  of  a  line  lying  in  the  plane 
uUu',  Fig.  5.  Find  d'ri,  and  const,  a  line  DC  lying  in 
uUu',  making  60°  with  DN.  Give  Z  c'd'n'.  (Two  solu- 
tions.) 

*I94-  Take  a'm'  as  the  V  proj.  of  a  line  lying  in  sSs'. 
Fig.  6.  Find  am,  and  const,  a  line  AC  lying  in  sSs'  and 
making  45°  with  AM.  Give  Z  cam.  (Two  solutions.) 

195.  Find  the  shortest  line  OQ  that  can  connect  the 
lines  MD  and  BR,  Fig.  4.     Give  its  length.     Const,  double 
size.     (The  pt.  R  is  in  the  G.  L.) 

196.  Find  the   shortest  line,   FJ,   connecting  the  lines 
GO  and  MD,  Fig.  8.     Give  its  length. 

197.  Find  the  shortest  line,  FJ,  that  can  be  drawn  bet. 
the  lines  ME  and  DN,  Fig.  8.     Give  its  length. 

198.  Find  the  shortest  line,  OQ,  connecting  the  lines 
AB  and  EC,  Fig.  4.     Give  its  length. 

199.  Find  the   shortest   line,  OQ,   concerting  the  lines 
AE  and  MD,  Fig.  4.     Give  its  length. 

200.  Find  the  shortest  line,  OQ,  connecting  CD  and 
EM,  Fig.  4.     Give  its  length. 


MISCELLANEOUS  PROBLEMS  ON  THE  POINT,  LINE 
AND  PLANE,  FOR  ORIGINAL  SOLUTIONS 

20 T.  The  V  proj.  of  a  line  1  to  DN,  Fig.  5,  is  q'd'. 
Find  the  H  proj. 

202.  civ  is  the  H  proj.  of  a  line  1  to  CE,  Fig.  6. 
Find  c'w'. 


*In  solving  Prob's   193  and   194,  no  attention  should  be  given 
to  d'n'  or  am  as  given  in  the  Fig's. 


DESCRIPTIVE  GEOMETRY  25 

203.  In  the  above  problem,  taking  CE  and  CW  as  two 
sides  of  a  rectangle,  complete  the  rectangle  ECWF  and 
obtain  by  points  the  projs.  of  the  inscribed  ellipse. 

204.  Obtain   by   points    the    projs.    of    the    parabola 
inscribed  in  the  above  rectangle. 

205.  Show   the   projs.   of   the   circle   inscribed   in  the 
triangle  ABD,  Fig.  8. 

206.  Show  the  projs.  of  the  circle  circumscribed  about 
the  triangle  ABD,  Fig.  8. 

207.  .  Thru   B,   Fig.  4,  const,  a  line,   BF,  making  20° 
with  H  and  40°   with  V. 

208.  Thru  N,  Fig.  5,  const,  a  line  NK  making  22^° 
with  H  and  30°  with  V. 

209.  A  pt.  M  is  in  the  ist  quadrant,  1.5"  from  H,  and 
2"  from  V.     A  pt.  N  is  in  the  ist  quadrant  0.5"  from  H 
and  3"  from  V.     The  length  of  the  line  MN  is  4".     Deter- 
mine the  projs.  of  the  line  MN. 

210.  A  rod  2}^  feet  long  is  suspended  horizontally  by 
vertical  threads  3  feet  long  attached  to  its  ends.     How  far 
will  the  rod  be  raised  by  turning  it  thru  90°  ? 

211.  How  far  will  the  above  rod  be  raised  by  turning 
it  thru  60°  ? 

212.  How  far  will  the  above  rod  be  raised  by  turning 
it  thru  120°  ? 

213.  The  H  trace  of  a  plane  tTt'  makes  an   Z   of  30° 
with  the  G.  L.     The  plane  makes  an    Z    of  45°   with  V. 
What    Z    does  the  V  trace  make  with  the  G.  L.  ? 

214.  The  V  trace  of  a  plane,  sSs',  makes  45°  with  the 
G.  L.    The  plane  makes  60°  with  H.     What  is  the  tangent 
of  the  Z  bet.  the  H  trace  and  the  G.  L.  ? 

215.  The  H  trace  of  a  plane,  tTt',  makes  60°  with  the 
G.  L.    The  Z  bet.  the  two  traces  in  space  is  75°.    What  Z 
does  the  V  trace  make  with  the  G.  L.? 

216.  The  H  trace  of  a  plane  sSs'  makes  an  Z  of  671/2° 
with  the  G.  L.    The  Z  bet.  the  two  traces  in  space  is  105° 
•What  is  the  tang,  of  the  Z  bet.  the  V  trace  and  the  G.  L. ? 


26  PROBLEMS    IN 

217.  A  pt.  Q  lies  in  the  plane  sSs',  Fig.  6.    Its  position 
when  developed  into  H  is  at  d.     Determine  the  two  projs. 
of  the  pt. 

218.  The  pt.  N,  Fig.  7,  lies  in  a  plane  kKk'.    Its  devel- 
oped position  in  H  is  at  d.     Const,  the  traces  of  the  plane 
kKk'.    The  H  trace  passes  betzveen  n  and  d. 

*2ic).  Take  b  as  the  H  proj.  of  a  pt.  lying  in  rRr',  Fig. 
8.  Thru  this  pt.  const,  a  line,  BK,  making  an  Z  of  30° 
with  H,  and  lying  in  the  plane. 

220.  Const,  a  line,  CE,  in  the  H  plane,  making  an   Z 
of  60°  with  DN,  Fig.  7.     Give  Z  bet.  CE  and  dn. 

221.  Given  bd,  Fig.  8,  as  the  H  trace  of  a  plane  kKk'. 
and  the  pt.  G,  1/4"  dist.  from  the  plane,  const,  the  V  trace. 
(Hint:     Thru  the  pt.  G  pass  a  plane  JL  to  the  given  trace 
and  revolve  it  into  the  corresponding  plane  of  proj.)     Two 
solutions. 

222.  Thru   B,   Fig.  8,  const,   a  line   BF  touching  the 
lines  GM  and  DO. 

223.  Thru  D,  Fig.  4,  const,  a  line  DN  touching  AE 
and  BC. 

224.  Thru  E,  Fig.  6,  pass  a  plane,  kKk',  ||  to  BM  and 
1  to  sSs'.     Give  dist.  KS. 

225.  The  line  OQ,  Fig.  7,  lies  in  uUu'.     Const,  its  H 
proj.,  and  thru  OQ  pass  a  plane  kKk'  making  an  Z  of  45° 
with  uUu'.      (Two  solutions.) 

226.  Thru  CD,  Fig.  8,  pass  a  plane  kKk',  making  an 
Z   of  45°  with  rRr'.   (  Two  solutions.) 

227.  Thru  D,  Fig.  5,  pass  a  line  DA  making  an   Z   of 
60°  with  rRr'  and  cutting  the  line  OQ,  which  lies  in  rRr'. 

228.  A  ray  of  light,  ND,  Fig.  5,  proceeding  from  the 
point  N  strikes  the  plane  uUu'.     Show  the  pts.  i,  2,  3  and 
4  where  it  is  successively  reflected  from  uUu'  an'd  the  three 
planes  of  proj.     Let  the  P  plane    pass    thru*  N.     Const, 
double  size. 


*In  Prob.  219,  no  attention  should  be  paid  to  the  V  proj.  of  B, 
as  given  in  the  figure. 


DESCRIPTIVE   GEOMETRY  2J 

229.  Of  a  plane  pentagon  BADGN,  suppose  we  have 
given  badgu  and  b'a'd',  Fig.  8.  Find  the  pts.  g'  and  n',  and 
the  true  figure,  without  constructing  the  plane  of  the  pen- 
tagon. (No  attention  should  be  paid  to  the  V  projs.  of  G 
and  N  as  given  in  the  figure.) 

^230.  Two  sides  of  a  plane  rectangle  are  horizontally 
projected  in  ab  and  ad,  Fig.  8.  A  lies  in  H,  and  the  true 
length  of  AB  is  2".  Const,  the  V  proj.  of  the  rectangle. 

231.  Pass  a  line  FK,  ||  to  AB,  Fig.  8,  and  cutting  DG 
and  WO. 

232.  Const,    the   projs.    of   a   regular   hexagon   whose 
center  is  D,  Fig.  8,  and  one  of  whose  sides  lies  in  AB. 

233.  Const,  a  line,  FJ,  lying  in  the  plane  sSs',  Fig.  8, 
and  1  to  DG. 

234.  Assume  a  2"  cube  in  ist  quadrant  with  faces  ||  to, 
and  dist.  1^/4"  from  H  and  V.     Required  the  projs.  of  the 
solid  formed  by  passing  planes  thru  the   12  edges  of  the 
cube,  each  1  to  the  diagonal  plane  in  which  the  edge  lies. 

235.  Bisect  the    Z    bet.  the  planes  rRr'  and  sSs',  Fig, 
6,  by  a  plane,  kKk'.     Draw  double  size. 

236.  Find  a  point,  N,  in  the  line  SB,  Fig.  4,  equally 
dist.  from  the  planes  gGg'  and  rRr'. 

237.  Thru  A,  Fig.  4,  pass  a  plane  sSs',  making  an   l_ 
of  75°  with  H  and  45°  with  V.     (Four  solutions.) 

238.  Thru  E,  Fig.  6,  pass  a  plane  tTt',  making  an   Z 
of  60°  with  H  and  67^°  with  V.     (Four  solutions.) 

239.  Const,    the   three   projs.   of   a  regular   hexagonal 
prism  3^/2"  high,  each  side  of  whose  base  is   i",  the  base 
making  an  Z   of  30°  with  H  and  75°  with  V,  and  one  edge 
of  the  base  being  ||  to  H ;  the  center,  C,  being  %"  from  H 
and   i1/?"  from  V. 

240.  Const,  the  projs.  of  a  i1/*"  cube  with  one  face  in 
H  and  one  face  making  an   Z   of  30°  with  V. 


*In   Prob.  230,  no  attention  should  be  paid  to  a',  b'  or  d'  as 
given  in  the  figure. 


28  PROBLEMS    IN 

241.  Const,  the  projs.  of  a  \y2"  cube,  one  edge  in  H 
making  30°   with  the  G.  L.  and  one  face  inclined  30°  to 
H  on  side  away  from  the  G.  L. 

242.  Const,   the  projs.   of  a     il/2"    cube,    one    "body 
diagonal"  being  1  to  H  and  the  H  proj.  of  one  edge  being 
1  to  the  G.  L. 

243.  A  cube  lies  with  one  of  its  faces  in  ullu',  Fig.  5. 
The  V  proj.  of  the  lowest  edge  is  d'n'.     Const,  the  three 
projs.  of  the  cube,  taking  the  P  plane  thru  W.     Const,  to 
scale   i^"=i". 

244.  Const,  the  projs.  of  a  cone  of  revolution  whose 
base  lies  in  rRr',  Fig.  8,  whose  vertex  is  at  A,  and  whose 
elements  make  an     Z    of    22^2°     with  the  axis.     Const 
double  size. 

245.  Const,  the  projs.  of  a  cone  of  revolution  whose 
vertex  is  at  A,  Fig.  8,  whose  base  has  a  dia.  of  i^"  and  lies 
in  the  plane  rRr'.     Const,  double  size. 

246.  Rotate  the  pt.  D,  Fig.  8,  about  the  line  AW,  and 
find  the  points  where  it  passes  thru  H,  V  and  P.     (Take 
P  thru  the  pt.  B.)     Also  show  the  position  of  D  after  it 
has  rotated  thru  90°   from  its  original  position. 

247.  Rotate  AB,  Fig.  8,  thru  90°   about  CD.     Show 
its  new  projs.     (Two  solutions.) 

248.  Const,  a  line,  FJ,  ||  to  rRr',  Fig.  8,  at  a  dist.  i" 
from  it,  and  cutting  the  lines  AB  and  CD.    (Two  solutions.) 

249.  Thru  D,  Fig.  4,  pass  a  line,  DF,  1  to  MD  and 
||  to  rRr'. 

250.  Pass  a  plane,  sSs',  twice  as  far  from  S  as  from 
D,  M  and  C,  Fig.  4.      (Two  solutions.) 

251.  Pass  seven  planes,  each  equally  dist.  from  S,  D, 
M  and  C,  Fig.  4. 

252.  Pass  a  plane,  tTt',  twice  as  far  from  S  and  M  as 
from  D  and  C,  Fig.  4.     (Two  solutions.) 

253.  Pass  a  plane,  sSs',  ||  to  both  AB  and  CD,  Fig;  8, 
and  equally  dist.  from  each. 

254.  Pass  a  plane,  tTt',  ||  to  both  AB  and  CD,  Fig.  8, 


DESCRIPTIVE  GEOMETRY  29 

and  twice  as  far  from  AB  as  from  CD.     (Two  solutions.) 

255.  The  point  G,  Fig.  8,  lies  in  a  plane  kKk'  whose 
H  trace  is  bd.     (a)   Find  the  V  trace,  giving  the    Z   bet. 
Kk'  and  the  G.  L.     (b)  Const,  the  projs.  of  a  sphere  of  i" 
radius,  tang,  to  kKk'  at  the  pt.  G.     (Two  solutions.) 

256.  Given  WNE,  Fig.  8,  as  the  base  of  a  triangular 
pyramid   lying  in   H,   and   the   lateral   edges,   WF=ij4"> 
NF=i>£"  and  EF— i",  const,  the  projs.  of  the  pyramid 
double  size.     Give  height  of  F  above  H. 

257.  Given  WEB,  Fig.  8,  as  the  base  of  a  triangular 
pyramid  lying  in  H,  and  the  lateral  edges,  WF=2" ;  EF= 
i^4";  BF=2j4",  const,  the  projs.  of  the  pyramid  double 
size.     Give  height  of  F  above  H. 

258.  Given  BWN,  Fig.  8,  as  the  base  of  a  triangular 
pyramid  lying  in  H,   and    the    lateral    edges,   BF=2^"; 
WF=i}^",  NF=2",  const,  the  projs.  of  the  pyramid.    Give 
height  of  F  above  H. 

259.  The  base  of  a  triangular  pyramid  has  its  sides, 
BC=3",   BD=2.s",   CD=2.25".     The    lateral    edges    are, 
AB=2%",    AC— 2$/s",    AD=2y4".     Const,  the    pyramid, 
giving  its  altitude. 

260.  Circumscribe  a  sphere  about  the  pyramid  in  the 
preceding  problem.     Give  radius. 

261.  Circumscribe  a  sphere  about  the  triangular  pyra- 
mid D— BWN,  Fig.  8.    Give  (a)  dist.  of  center  above  H, 
(b)   radius  of  circumscribed  sphere. 

262.  Circumscribe  a  sphere  about  the  triangular  pyra- 
mid O — WEN,  Fig.  8,     Give  (a)  dist,  of  center  above  H, 
(b)    radius  of  circumscribed   sphere.     Const,   double   size. 

263.  Pass  a  sphere  thru  the  four  pts.  B,  D,  G  and  W, 
Fig.  8.    Give  radius  of  sphere  and  dist.  of  center  above  H. 

264.  Inscribe  a  sphere  in  the  triangular  pyramid  D — 
BWN,  Fig.  8.    Give  radius  of  sphere.     Const,  double  size. 

265.  Inscribe  a  sphere  in  the  triangular  pyramid  of 
Prob.  259.     Const,  double  size,  and  give  radius  of  sphere. 


30  PROBLEMS   IN 

TRIHEDRAL   ANGLES— GRAPHICAL    SOLUTION    OF 
SPHERICAL  TRIANGLES 

In  the  following  problems,  let  A,  B  and  C  represent  the 
dihedral  angles  (or  spherical  angles),  and  a,  b  and  c  the 
face  angles  (or  sides  of  the  spherical  triangle)  opposite 
A,  B  and  C  respectively. 

266.  Given  £=63°  1 5';  c=tf^2f\  O=55°53';  find  A, 
B  and  a. 

267.  Given  a=79°i';  b=&2°  i/;  A=82°9';  find  B,  C 
and  c. 

268.  Given  0=64° 47' ;  c=48°3';  C=54°8' ;  find  A,  B 
and  b. 

269.  Given   0=38°  ;  6=42°;   B=s8°53/;    find    A,    C 
and  c. 

270.  Given    B=82°;    0=67°;    /?=73°48';    find    a,  c 
and  A. 

271.  Given  A=59°4' ;  B=88°i2/;  a=so°2';  find  b,  c 
and  C. 

272.  Given  A=5i°s8';  C=83°55/;  c=$i°  ;  find  a,  b 
and  B. 

273.  Given   A=73°4/ ;   B=54°8' ;   a=6i°47'\   find   b, 
c  and  C. 

274.  Given   a-=  6i°47/;   ^=43°3r;   C=8o°2o';   find   A, 
B  and  c. 

275.  Given   0=79° i';   ^=82°  17';   C=45°44V;   find   A, 
B  and  c. 

276.  Given   /?=63°i5';   c=47°42r ;   A=59°4/;   find    B, 
C  and  a. 

277.  Given   0=64° 47' ;   £=6i°47';   C=54°8/ ;   find   A, 
B  and  c. 

278.  Given   6=90°;   C=45°44' ;  a=79°l''>  find   A?   b 
and  c. 

279.  Given    B=88°i2';   C=55°53/;   a^=5o°2; ;   find   b, 
c  and  A. 

280.  Given  A=83°55':  6=51°  58';  ^=-42°:  find  a,  b 
and  C. 


DESCRIPTIVE   GEOMETRY  31 

281.  Given   B=7304/ ;   C=54°8' ;   0=64° 47';  find   b, 
c  and  A. 

282.  Given  0=51° ;  £=38°  ;  ^=42°  ;  find  A,  B  and  C. 

283.  Given  #=63°  1 2';   b=6$°46' ;  c=73°48';  find  A, 
P.  and  G. 

284.  Given    a=47°42' ;    b=$o°2f ;    ^=63°  15';    find   A, 
P>  and  C. 

285.  Given  a=48°3';  b=64°47';  c=6i°4f\  find  A,  B 
and  C. 

286.  Given  A=83055';  6=51  "58';  C=58°53';  find  a, 
b  and  c. 

287.  Given  A=54°8' ;   B=8o°2o';   C=73°47';  find  a, 
b  and  c. 

288.  Given    A=45°44' ;    B=82°9' ;    C=9O°  ;    find    a, 
b  and  c. 

289.  Given   A=59°4' ;   B=88°i2';   C=55°53';  find  a, 
b  and  c. 


DETERMINATION   OF   SKEW  ANGLES  IN    ROOF 
STRUCTURES 

An  application  of  Descriptive  Geometry  to  the  solution 
of  practical  problems  involving  the  right  line  and  plane  is 
shown  in  the  determination  of  skew  angles  in  beam  con- 
nections for  roofs  or  other  structures.  The  presentation  of 
the  subject  here  given  is  a  modification  of  that  given  by 
Prof.  C.  G.  Wrentmore,  and  published  in  The  Michigan 
Technic  of  1903.  For  the  solution  of  the  same  problem 
by  spherical  trigonometry,  see  Mr.  C.  A.  P.  Turner's  article 
in  Eng.  News  of  Feb.  15  and  22,  1900. 

Since  the  graphical  method  insures  against  the  placing 
of  two  bodies  in  the  same  place  at  the  same  time,  it  is 
considered  preferable  to  the  method  of  computation.  The 
experience  of  those  who  have  tried  both  methods  has  shown 
that  the  graphical  method  presents  less  probability  of  error 
and  greater  probability  of  all  clearances  being  provided  for, 

imJAS 

OF  THF 

UNIVERSITY 

OF 


32  PROBLEMS  IN 

than  the  method  by  computation;  and  where  the  principal 
dimensions  are  checked  by  computation,  a  drawing  to 
quarter  size  will  give  all  secondary  dimensions  with  suffi- 
cient accuracy  for  all  cases,  both  for  angular  and  linear 
measurements. 

Drawings  for  this  purpose,  however,  must  be  done  with 
exactness ;  and  the  draftsman  must  know  his  own  limita- 
tions in  the  matter  of  accurate  work  before  he  can  deter- 
mine how  far  his  graphical  solutions  are  trustworthy. 

Fig.  9  is  a  perspective  view  of  several  roofs  forming 
hips  and  valleys.  In  roofs  R,  R'  and  R",  horizontal  lines 
run  at  right  angles ;  but  in  roofs  R",  R'",  etc.,  horizontal 
lines  run  at  an  angle  of  135°. 

Fig.  10  is  a  descriptive  drawing  for  the  solution  of 
angles  connected  with  roofs  R  and  R'.  The  planes  of  pro- 


Tig.  9. 

jection  used  are,  the  horizontal  plane,  two  vertical  planes 
respectively  1  to  the  H  traces  of  the  roof  planes,  and 
another  vertical  plane  through  the  hip  MP.  We  have  then 
three  ground  lines:  XX',  op"  and  mp.  The  V  plane  that 
is  used  for  any  particular  solution  is  theoretically  imma- 
terial;  but  practically  much  time  and  labor  may  be  saved 
by  choosing  the  best  one. 

In  these  problems  the  pitch  of  each  roof  (height  divided 


DESCRIPTIVE   GEOMETRY  33 

by  span)  will  be  given,  and  the  following  angles  will  be 
worked  out,  the  same  general  method  being  used  whether 
the  roofs  form  a  hip  or  a  valley,  or  whether  the  eaves  run 
at  right  angles  or  otherwise. 

A  is  the  angle  bet.  plane  of  roof  and  the  H  plane 

We  choose  the  pt.  p  (Fig.  10)  in  any  convenient  part 
of  the  sheet,  and  draw  the  H  traces  pp'  and  pp"  of  the  two 
roofs.  It  is  evident  that  the  tangent  of  the  Z  A  is  double 
the  pitch.  If  then  the  pitch  of  the  roof  R  is  given  as  J4, 
the  Z  A  will  have  a  tangent  of  J^.  We  lay  off  two  units 
to  the  right  of  p'  and  one  unit  up,  giving  us  the  point  m\ 
The  V  trace  of  the  roof  plane  R  is  p'm',  and  the  required 
Z  is  A.  Similarly  we  obtain  A'  from  the  given  pitch  of 
R'  (in  this  case  1/3). 

B  is  the  angle  bet.  the  H  trace  of  the  roof  plane, 
and  the  H  trace  of  the  hip-web  plane,  or  valley-web 
plane 

Assume  any  pt.  m"  in  the  roof  line  p"m"  at  a  height 
nm"  above  H,  and  find  the  pt.  m'  in  the  other  roof  line  at 
the  same  height,  by  taking  ym'=nm".  Draw  lines  thru  y 
and  M  ||  to  the  eave  lines  pp'  and  pp"  respectively.  They 
meet  in  w;  and  pm  is  the  H  proj.  of  the  hip.  It  is  also 
the  H  trace  of  the  hip-web  plane,  and  p'pm  is  the  Z  B  for 
roof  R ;  and  p"pm  is  the  Z  B'  for  the  roof  R'. 

C  is  the  angle  bet.  the  hip  (or  valley)  rafter,  and 
the  H  plane 

Revolve  the  H  projecting  plane  of  the  hip  into  H.  The 
pt.  M  will  fall  at  m"',  whose  dist.  from  m  is  equal  to  ym'. 
The  revolved  position  of  the  hip  is  pm'",  and  mpiri"  is 
the  Z  C. 


34 


rig.  10. 


DESCRIPTIVE   GEOMETRY  35 

D  is  the  angle  in  the  roof  plane  bet.  a  main  rafter 
and  the  hip  rafter  (or  valley  rafter) 

Revolve  the  roof  plane  about  its  H  trace  pp'  into  H. 
Any  pt.  in  the  hip  line,  as  S,  will  fall  into  the  position  s'", 
and  the  hip  will  fall  into  the  position  ps"'.  Any  main  rafter 
is  1  to  pp'  and  falls  ||  to  pp" .  Hence  s'"pp"  is  the  value 
of  D.  Proceed  in  a  similar  manner  for  the  Z  D'. 

E  is  the  angle  bet.  a  vertical  line  and  the  trace 
of  the  purlin  web  upon  the  hip  web  plane 

The  plane  of  the  purlin  web  is  taken  normal  to  the 
plane  of  the  roof,  its  H  trace  being  my;  its  V  trace  yr'  and 
its  trace  on  the  hip  web  plane  mr'" .  This  trace  makes  with 
the  vertical  line  rr'"  the  Z  E  required.  Similarly  for 
the  Z  E'. 

F  is  the  angle  bet.  the  roof  plane  and  plane  of 
back  of  hip  (or  valley) 

These  two  planes  intersect  in  the  hip  line  PM.  The 
1 1  trace  of  the  back  of  hip  is  pz.  Pass  a  plane  tTt'"  normal 
to  the  hip.  Its  H  trace  is  Tt,  normal  to  pin.  Its  trace  on 
the  hip  web  plane  is  Tt"',  normal  to  pin"'.  It  cuts  from 
the  back  of  hip  a  line  normal  to  pm,  and  from  the  plane  of 
R  a  line  which,  when  revolved  about  Tt  into  H,  falls  into 
the  position  at.  The  Z  F  is  therefore  the  Z  bet.  at  and 
a  line  normal  to  pm,  or  the  Z  Tta. 

G  is  the  angle  in  purlin-web  bet.  a  normal  to 
center  line  of  purlin,  and  the  trace  of  hip-web  on  pur- 
lin-web 

The  intersection  of  hip  and  purlin  webs  is  horizontally 
projected  in  mr.  Revolving  the  purlin  web  plane  about  its 
H  trace,  my,  the  point  R  falls  at  r1?  and  the  line  mr  takes 
the  position  nir}.  The  required  Z  is  mr^r. 


36  PROBLEMS    IN 

H  is  complement  of  angle  bet.  purlin-web  and 
hip-web  planes 

A  plane  _L  to  the  intersection  (MR)  of  purlin-web  and 
hip-web  is  assumed  at  uUu"'.  This  cuts  from  the  purlin- 
web  a  line  which  revolves  into  H  at  w«1?  and  from  the 
hip-web  a  line  which  revolves  into  H  in  Uiij.  The  Z  bet 
the  planes  is  therefore  shown  at  m^U,  and  its  complement 
at  i^uU. 

I  is  the  angle  in  back  of  hip  bet.  a  line  normal 
to  hip-web  and  the  trace  of  purlin-web  on  back  of  hip 

The  H  trace  of  back  of  hip  is  ps.  This  plane  is  cut  by 
the  purlin  web  in  the  line  horizontally  projected  at  rz. 
When  this  line  is  revolved  about  pz,  it  falls  at  r.2z.  A  line 
normal  to  the  hip  web  will  revolve  into  H  in  a  parallel  to 
/>£.  The  Z  p£J'n  is  therefore  the  Z  I  required. 

J  is  complement  of  angle  in  purlin-web  bet.  traces 
of  hip-web  and  back  of  hip 

The  line  rz  revolved  about  yz  into  H  falls  at  r^z.  The 
revolved  position  of  mr,  as  already  found,  is  mr^.  The  Z 
bet.  these  lines,  mr^z,  is  therefore  the  complement  of  the 
Z  J,  which  in  this  case  is  a  negative  angle. 

K  is  angle  bet.  back  of  hip  and  purlin  web 

The  line  of  intersection  of  the  two  planes  is  RZ.  The 
II  projecting  plane  of  this  line  is  revolved  about  its  H 
trace,  rz,  into  H  ;  the  line  RZ  then  falling  at  r^z.  A  plane 
normal  to  both  planes  will  have  its  H  trace  at  be,  JL  to  rz ; 
and  its  trace  upon  the  H  projecting  plane  of  RZ  is  shown 
in  revolved  position  at  de.  If  the  plane  be  be  now  revolved 
about  be  into  H,  the  pt.  where  it  cuts  the  line  RZ  falls  at  g, 
and  the  Z  bet.  the  planes  is  shown  at  bge. 


DESCRIPTIVE  GEOMETRY  37 

290.  Determine  graphically   the   eleven    Zs,   A'  to   K' 
inclusive,  for  the  roof  R',  when  the  pitch  of  R  is  %,  and 
that  of  R'  is  1/3. 

291.  Determine  the   Zs  A'  to  K'  for  the  roof  R'  when 
pitch   R'=i/3,   pitch   R"=y2,   and   the   two   roofs   form   a 
valley  with  eaves  at  rt.  Z s. 

292.  Determine  the    Zs  A"  to  K"  for  the  roof  R"  in 
the  above  case. 

293.  Determine  the  Z s  A"  to  K"  for  the  roofs  R"  and 
R"'  when  each  has  a  pitch  of  */2,  and  the  eaves  form  an 
L   of  135°. 

294.  Determine  the  Z s  A"  to  K"  for  the  roof  R"  when 
pitch  R"=i/3  and  pitch  R'"=2/5.     Roofs  form  a  hip,  and 
caves  run  at  an   Z   of  120°. 

295.  Determine  the   Zs  A'"  to  K'"  for  the  roof  R"'  in 
the  above  case. 

296.  Determine  the   Z s  A  to  K  f or  the  roof  R,  when 
pitch  R=2/5,  pitch  R'=^,   roofs   form  a  hip  and  eaves 
form  a  rt.    Z  . 

297.  Determine  the    Z8  A'  to  K'  for  the  roof  R'  in 
the  above  case. 

298.  Determine  the  Zs  A'  to  K'  for  the  roof  R',  when 
pitch  R'=*4,  pitch  R"=2/5,  roofs  form  a  valley  and  eaves 
run  at  rt.    Zs. 

299.  Determine  the    Zs  A"  to  K"  for  the  roof  R"  in 
the  above  case.  

REPRESENTATION  OF  SURFACES  AND  SPACE 
CURVES 

300.  Const,  the  H  and  V  projs.  of  one  complete  con- 
volution of  a  helix  with  axis  _L  to  H.     Generating  point 
moves  4"  in  direction  of  axis  for  each  revolution  about  the 
axis.     Dist.  of  generating  pt.  from  axis  i}/2". 

301.  A  right  circular  cone  of  3%"  alt.,  and  dia.  of  base 
3".  is  resting  with  its  base  in  H.    Another  rt.  circular  cone, 
with  equal  elements,  and  dia.  of  base  ij^",  has  its  vertex 


38  PROBLEMS   IN 

coincident  with  that  of  the  first  cone,  and  an  element  in 
common  with  it.  Const,  the  H  and  V  projs.  of  the  spher- 
ical epicycloid  generated  by  a  pt.  in  the  circumference  of 
the  base  of  the  second  cone  as  it  rolls  upon  the  first. 

302.  Const,  the  H  and  V  projs.  of  a  cone  of  revolu- 
tion  with   base   lying  in   H,   and   vertex   in    ist   quadrant. 
Dia.  of  base,  2" ;  alt.  of  cone,  3".     Assume  an  element,  SA. 
of  the  cone,  and  show  the  inclination  of  the  elements  to  H. 

303.  Const,  the  H,  V  and  P  projs.  of  an  oblique  cone 
with   an  elliptical  base   lying  in   H,   and   with  axis   in    ist 
quadrant  oblique  to  H,  V  and  P.     Assume  a  point,  A,  on 
the  surface. 

304.  A  right  circular  cone  with  axis  ON,  Fig.  8,  base 
in  H,  and  dia.  of  base  i/4",  is  in  contact  with  an  equal  cone 
lying  on  an  element  in  H,  and  with  its  vertex  at  E.  Const, 
the  projs.  of  the  two  cones. 

305.  Const,  the  projs.  of  a  cyl.  of  revolution,  base  in 
H,  axis  in  V,  dia.  of  base  i",  elements  inclined  45°  to  H. 

306.  Const,  the  projs.  of  an  oblique  cyl.  with  circular 
base  lying  in  H  ;  elements  ||  to  V,  and  inclined  60°  to  H ; 
dia.  of  base,   1/4"-     Assume  an  element  AB  and  a  pt.  C, 
on  the  surface. 

307.  Assuming  the  ellipse  in  Fig.    12,  as  the  base  of 
a  cyl.   of  revolution   with  elements   ||   to  V,  const,   the  V 
proj.  of  the  cyl.  and  give  the  inclination  of  the  elements 
to  H. 

308.  Assume  a  point,  A,  on  the  surface  of  a  helicoid 
whose  elements  are  inclined  at  an  angle  of  60°  to  the  axis. 
Generating  pt.  of  helix  moves  4"  in  direction  of  axis  for 
each   revolution   about   the   axis.      Dist.   of   generating  pt. 
from  axis,  1^4". 

309.  Given  CD  and  GM,  Fig.  8,  as  the  directrices  of 
an  hyperbolic  paraboloid,  with  rRr'  as  plane  directer,  as- 
sume an  element  of  the  surface  through  G.     Assume  also 
a  point  F  on  that  element. 

310.  Given  GO  and  DW,  Fig.  8,  as  the  directrices  of 


DESCRIPTIVE   GEOMETRY  39 

an  hyperbolic  paraboloid,  the  plane  directer  being  the  H 
plane,  const,  an  element  thru  D,  and,  one  thru  W,  and  as- 
sume a  point  F  on  the  first  element. 

311.  Given  CD  and  GM,  Fig.  8,  as  the  directrices  of 
an   hyperbolic   paraboloid,   and   DG   and   CM   as   two  ele- 
ments of  the  surface,  find  a  plane  directer  for  each  genera- 
tion. 

312.  Assume    a    hyperbolic    paraboloid    with    CD    and 
WO,  Fig.  8,  as  directrices,  and  CO  and  DW  as  two  ele- 
ments.   The  H  proj.  of  a  pt.  F  on  the  surface  is  iV%"  below 
S.     Find  /'  and  give  its  dist.  from  the  G.  L. 

313.  With  d,  Fig.  8,  as  center,  strike  an  arc  to  the  left 
with  radius  2*4" '.    With  d'  as  center  and  radius  2j4" '»  strike 
an  arc  to  the  left.     These  are  the  H  and  V  projs.  of  a 
curved  line.     With  c'  as  center  and  radius  3",  strike  an  arc 
to  the  right  and  below  the  G.  L.,  and  with  a'  as  center,  and 
3"  radius,  strike  an  arc  to  the  right  and  above  the  G.  L. 
These  are  the  H  and  V  projs.  of  another  curved  line.    Tak- 
ing these  as  the  directrices  of  a  warped  surface  with  rRr' 
as  the  plane  directer,  const,  the  projs.  of  an  element  of  the 
warped  surface  thru  a  pt.  in  the  left  hand  curve  i"  above  H. 

314.  Const,  the  two  projs.  of  an  hyperboloid  of  rev- 
olution of  one  nappe,  with  ON,  Fig.  8,  as  axis,  O  the  cen- 
ter of  the  gorge  circle  whose  dia.  is   i",  and  elements  in- 
clined 60°  to  H.     Assume  a  point,  Q,  on  the  surface. 

315.  Const,  the  projs.  of  the  right  and  left  hand  ele- 
ments of  a  conoid  with  AD,  Fig.  8,  as  one  directrix,  and 
a  circle  in  H  with  center  at  B  and  il/2"  dia.,  as  the  other 
directrix ;  the  plane  directer  being  sSs'.     Assume  a  point 
Q  on  the  surface. 

316.  The   three   directrices   of  an   hyperboloid   of  one 
nappe  are  BD,  WO,  and  GN,  Fig.  8.     vShow  the  projs.  of 
an  element  thru  B,  and  of  one  thru  O. 

317.  Assume  a  point,  K,  on  a  sphere  with  center  at 
O,  Fig.  8,  and  dia.  iy2" . 

318.  A  prolate  ellipsoid   of  revolution  has  its  axis  in 


40  PROBLEMS    IN 

ON,  Fig.  8,  and  its  center  at  O.  The  axes  of  the  revolv- 
ing ellipse  are  2,y2"  and  iy2".  Show  the  projs.  of  the  ellip- 
soid, and  assume  a  point,  F,  on  the  surface. 

319.  Taking  the  helix  in  Prob.  300  as  the  directrix  of 
a  helical  convolute,  assume  an  element  of  the  surface  and 
a  point  K  on  the  surface.    Also  const,  the  trace  of  the  sur- 
face on  the  H  plane. 

320.  Assume  a  helix  with  axis  1  to  the  G.  L. ;  pitch 
3J4";  dia.  of  H  proj.  3^".     Taking  this  as  the  directrix 
of  a  helical  convolute,  assume  an  element  of  the  surface, 
and  a  point  on  that  element,  and  const,  a  portion  of  the 
trace  of  the  surface  on  the  H  plane. 


PROBLEMS  RELATING  TO  LINES  AND  PLANES 
TANGENT  TO  SINGLE  CURVED  SURFACES 

321.  Thru  A,  Fig.  n,  on  the  surface  of  the  left  hand 
cyl.,  pass  a  plane,  tTt',  tang,  to  the  cyl.     The  pt.  A  is  on 
the  side  nearest  the  V  plane.     Give   Z   RTt'. 

322.  The  pt.  B,  Fig.   n,  is  on  the  lower  side  of  the 
cyl.     Pass  a  plane  sSs',  thru  B,  tang,  to  the  cyl.     Give   Z 
RSs'. 

323.  The  pt.  E.  is  on  the  upper  side  of  the  right  hand 
cyl.  in  Fig.   n.     Pass  a  plane,  tTt',  thru  E,  tang,  to  the 
cyl.     Give   Z    RTt'. 

324.  Take  the  ellipse  in  Fig.  12  as  the  base  of  a  cyl. 
with  elements   ||  to  AB.     Pass  a  plane,  tTt',  thru  E  and 
tang,  to  the  cyl. ;  the  point  being  on  that  portion  of  the 
vSurface  of  the  cyl.  nearest  V.     Give  Z   RTt. 

325.  Take  the  left  hand  circle  in  Fig.   n  as  the  base 
of  a  cyl.  with  elements  ||  to  V  and  inclined  60°  to  H.    Pass 
a  plane  sSs'  thru  A  and  tang,  to  the  cyl. ;  the  pt.  A  being 
on  that  portion  of  the  cylindrical  surface  nearest  V.     Give 
Z   RSs'.     Draw  double  size. 

326.  Take  BN,  Fig.  12,  as  the  axis  of  a  cyl.  of  revo- 
lution  with   dia.    i".     Thru   C,   on  the  upper   side,  pass  a 


DESCRIPTIVE   GEOMETRY 


PROBLEMS   IN 


/         £       3-gy         ^j 


DESCRIPTIVE   GEOMETRY  43 

plane,  tTV,  tang,  to  the  cyl.    Give  dist.  of  each  trace  from 
G.  L. 

327.  Take  BN,  Fig.  12,  as  the  axis  of  a  cyl.  of  revo- 
lution with  dia.   1*4".     Thru  E,  on  the  side  furthest  from 
V,  pass  a  plane,  sSs',  tang,  to  the  cyl.     Give  dist.  of  each 
trace  from  G.  L. 

328.  The  pt.  A,  Fig.  12,  is  on  the  upper  surface  of  the 
left  hand  cyl.     Pass  a  plane  tTt'  thru  A  and  tang,  to  the 
cyl.     Give  Z   KTf. 

329.  If  the  elements  of  the  cyl.  K,  in  Fig.  12  are  in- 
clined 60°  to  H,  and  a  is  the  H  proj.  of  a  point  on  the 
lower  surface  of  the  cyl.,  find  a',  and  pass  a  plane  sSs', 
thru  A  and  tang,  to  the  cyl.     Give   Z   RSs'. 

330.  Thru  C,  Fig.  n,  pass  two  planes,  tTt'  and  sSs', 
tang,  to  the  left  hand  cyl.     Give   Z s  RTt'  and  RSs'. 

331.  Thru  G,  Fig.  n,  pass  two  planes,  sSs'  and  tTt', 
tang,  to  the  rt.  hand  cyl.    Give  Z s  KS/  and  KT/'. 

332.  Thru  M,  Fig.  n,  pass  a  plane,  tTt',  tang,  to  the 
rt.  hand  cyl.,  on  the  side  furthest  from  the  V  plane.     Give 
Z  bet.  traces  on  drawing. 

333.  Thru  D,  Fig.  n,  pass  a  plane,  sSs',  tang,  to  the 
left  hand  cyl.  on  the  side  nearest  the  V  plane.    Give  Z  bet. 
traces  on  drawing. 

334.  Thru  Q,  Fig.  12,  pass  a  plane  sSs'  tang,  to  a  cyl. 
of  revolution  whose  axis  is  BN,  and  whose  dia.  is   i1/^' '. 
Give  dist.  bet.  traces  on  drawing. 

335.  Thru  A,  Fig.   12,  pass  a  plane,  sSs',  tang,  to  a 
cyl.   of   revolution   whose   axis   is   BN,   and   whose   dia.   is 
i".     Let  the  plane  be  passed  on  the  lower  side  of  the  cyl. 
Give  dist.  bet.  traces  on  drawing. 

336.  Thru  D,  Fig.   12,  pass  a  plane,  tTt',  tang,  to  a 
cyl.  of  revolution  whose  axis  is  BN,  and  whose  dia.  is  i". 
Let  the  plane  be  passed  on  the  lower  side  of  the  cyl.     Give 
dist.  bet.  traces  on  drawing. 

337.  Let  the  ellipse  in  Fig.   12,  be  taken  as  the  base 
of  a  cyl.  whose  right  section  is  a  circle  and  whose  elements 


44  PROBLEMS   IN 

run  ||  to  V.  Const,  the  projs.  of  the  cyl.,  and  pass  two 
planes,  sSs'  and  tTt',  tang,  to  it  thru  D.  Give  distances 
RS  and  RT. 

338.  Thru  Q,  Fig.  12,  pass  a  plane,  sSs',  tang,  to  the 
left  hand  cyl.  on  the  side  nearest  the  V  plane.     The  ele- 
ments of  the  cyl.  make  an    Z    of  60°   with  H.     Give    Z 
RSs'. 

339.  Pass  a  plane,  tTt',  tang,  to  the  left  hand  cyl.  of 
Fig.    ii   and   ||   to  the  line  CD.     Give    Z    bet.   traces   on 
drawing. 

340.  Pass  a  plane,  sSs',  tang,  to  the  rt.  hand  cyl.  of 
Fig.    ii   and   ||   to  the  line  GM.     Give    Z    bet.  traces  on 
drawing. 

341.  Pass  a  plane,  sSs',  tang,  to  the  left  hand  cyl.  of 
Fig.    n,  and   ||   to  the  line  DG.     Give    Z    bet.  traces  on 
drawing. 

342.  Pass  a  plane,  tTt',  tang,  to  the  left  hand  cyl.  of 
Fig.   n,  and   ||   to  the  line  DM.     Give    Z    bet.  traces  on 
drawing. 

343.  Pass  a  plane,  sSs',  tang,  to  left  hand  cyl.  of  Fig. 
12,  and  ||  to  the  line  AB.     The  axis  of  the  cyl.  makes  an 

/   of  60°  with  H. 

344.  Pass  a  plane,  tTt',  tang,  to  the  rt.  hand  cyl.  of 
Fig.  ii,  and  ||  to  the  G.  L.    Give  dist.  of  Tt'  from  the  G.  L. 

345.  Assume  a  rt.  circular  cyl.  whose  axis  is  BN,  Fig. 
12,  and  whose  dia.  is  i%" '.     Pass  a  plane,  sSs',  tang,  to  it 
and  ||  to  AB.     Give  dist.  bet.  traces  on  drawing. 

346.  Assume  a  rt.  circular  cyl.  with  axis  BN,  Fig.  12, 
and  dia.  \Y^ '.     Pass  a  plane,  sSs',  tang,  to  it  and  ||  to  OQ. 
Give  dist.  bet.  traces  on  drawing. 

347.  Assume  a  rt.  circular  cyl.  with  axis  BN,  Fig.  12, 
and  dia.  i".     Pass  a  plane  sSs'  tang,  to  it  and  ||  to  OB. 
Give  dist.  bet.  traces  on  drawing. 

348.  Thru   M,  on  the  lower  surface  of  the  rt.  hand 
cone  in  Fig.   12,  pass  a  plane,  sSs',  tang,  to  the  surface. 
Give   Z  KSs'.     Const,  double  size. 


DESCRIPTIVE  GEOMETRY  45 

349.  Take  o',  Fig.  12,  as  the  V  proj.  of  a  pt.  on  the 
surface  of  the  cone,  and  on  the  side  toward  V.     Pass  a 
plane,  tTt',  thru  this  pt.  tang,  to  the  cone.     Give   Z   RTt'. 
Const,  double  size.     (No  attention  should  be  paid  to  the 
H  proj.  of  O  as  given  in  the  figure.) 

350.  Assuming  Q,  Fig.    12,  as  the  vertex  of  a  cone 
with  the  ellipse  as  base,  pass  a  tang,  plane,  tTt',  thru  the 
pt.  (on  the  lower  surface  of  the  cone)  of  which  m  is  the 
H  proj.    Give  Z  KTt'. 

351.  Thru  E,  Fig.   12,  situated  on  the  surface  of  the 
cone  furthest  from  the  V  plane,  pass  a  plane,  tTt',  tang, 
to  the  cone.    Give  Z   KTt'.     Const,  double  size. 

352.  The  vertex  of  a  rt.  circular  cone  is  at  B,  Fig.  12, 
Axis  is  ||  to  the  G.  L.     Base  is  2^"  to  the  rt.  of  the  ver- 
tex, and  2"  in  dia.     Thru  C,  on  the  upper  surface  of  the 
cone,  pass  a  plane  sSs',  tang,  to  the  surface.     Give   Z   bet. 
traces  on  drawing. 

353.  Take  N,  Fig.  12,  as  the  vertex  of  a  cone  of  rev- 
olution with  axis  ||  to  the  G.  L.     The  base  is  i%"  in  dia. 
and  2*4"  to  the  left  of  N.     Thru  E,  on  surface  furthest 
from  V  plane,  pass  a  planet  sSs',  tang,  to  the  cone.     Give 
Z   bet.  traces  on  drawing. 

354.  Take  W,  Fig.  12,  as  the  vertex  of  a  cone  of  revo- 
lution with  axis  coincident  with  the  G.  L.     The  base  is 
2y2"  to  the  left  of  W,  and  2*4"  in  dia.     Thru  E,  a  pt.  on 
the  surface  in  ist  quadrant,  pass  a  plane  sSs',  tang,  to  the 
cone.    Give  Z  bet.  traces  on  drawnig. 

355.  Pass  two  planes,  sSs'  and  tTt',  thru  O,  Fig.   12, 
tang,  to  the  surface  of  the  left  hand  cone.     Give   Z 8  RTt' 
and  RSs'. 

356.  Thru  Q,  Fig.  12,  pass  a  plane,  tTt',  tang,  to  the 
surface  of  the  rt.  hand  cone  on  the  side  furthest  from  the 
V  plane.    Give  Z   RTt'. 

357.  Thru  B,  Fig.  12,  pass  a  plane,  tTt',  tang,  to  the 
right  hand  side  of  the  left  hand  cone.     Give   Z    RTt'. 

358.  Assuming  the  rt.  hand  circle,  Fig.  u,  as  the  base 


46  PROBLEMS   IN 

of  a  cone,  and  G  as  the  vertex,  pass  a  plane,  tTt',  thru  D, 
and  tang,  to  the  surface  of  the  cone  on  its  left  side.     Give 
/    KTt'. 

359.  Assume  C,  Fig.   n,  as  the  vertex  of  a  cone  of 
revolution,  with  axis  ||  to  the  G.  L.,  base  i*4"  in  dia.,  and 
2%"  to  the  right  of  C.     Pass  two  planes,  sSs'  and  tTt', 
thru  D  and  tang,  to  the  surface.     Give   Z s  KSs'  and  KTt'. 

360.  Assume  D,  Fig.   n,  as  the  vertex  of  a  cone  of 
revolution  with  axis  ||  to  the  G.  L.,  base  il/2ff  in  dia.  and 
21/4"  to  the  left  of  D.     Pass  two  planes,  sSs'  and  tTt',  thru 
C  and  tang,  to  the  surface.    Give  Z s  WSs'  and  WTt'. 

361.  Assume  R,  Fig.   n,  as  the  vertex  of  a  cone  of 
revolution  with  axis  lying  in  G.  L.,  base  i1/*"  in  dia.  and 
3"  to  the  left  of  R.     Pass  a  plane,  tTt',  thru  G  and  tang, 
to  the  upper  surface  of  the  cone.     Give   Z    bet.  traces  on 
drawing. 

362.  Pass   a  plane,   sSs',  tang,   to  the  surface  of  the 
left  hand  cone,  Fig.  12,  on  the  side  nearest  V,  and  ||  to  the 
line  AB.     Give   Z    RSs'. 

363.  Pass  a  plane,  sSs',  tang,  to  the  right  hand  cone 
in  Fig.  12  and  ||  to  the  line  OO,  on  the  side  furthest  from 
V  plane.     Give   Z   RSs'.     Const,  double  size. 

364.  Take  the  circle  in  Fig.  12,  as  the  base  of  a  cone, 
and  A  as  the  vertex,  and  pass  a  plane  sSs',  tang,  to  the 
surface  on  the  side  furthest  from  V,  and  ||  to  the  line  OO. 
Give  Z  bet.  G.  L.  and  V  trace.    Const,  double  size. 

365.  Assume  the  rt.  hand  circle  in  Fig.  n  as  the  base 
of  a  cone,  and  G  as  the  vertex.     Pass  two  planes,  sSs',  and 
tTt',  ||  to  CD  and  tang,  to  the  surface  of  the  cone.     Give 
Zs  KSs'  and  KTt'. 

366.  Assume   a    rt.    circular   cone   with   vertex   at    D, 
Fig.  n,  axis  ||  to  the  G.  L.,  base  il/\"  in  dia.  and  2l/2f>  to 
the  right  of  D.     Pass  two  planes  sSs',  and  tTt',  tang,  to 
the  cone,  and  ||  to  the  line  GM.    Give  Zs  RSs'  and  RTt'. 

367.  Assume  a  rt.  circular  cone  with  vertex  at  B,  Fig. 
12,  axis  ||  to  the  G.  L.,  base  1*4"  in  dia.  and  2l/2"  to  the 


DESCRIPTIVE:  GEOMETRY  47 

left  of  B.     Pass  two  planes,  sSs'  and  tTt',  tang,  to  the 
cone  and  ||  to  the  line  OQ.     Give   Zs  KSs'  and  KTt'. 

368.  Assume  a  rt.  circular  cone,  axis  in  the  G.  L.  dia 
of  base  =   iV^",  alt.  —  3",  vertex  on  left  side  of  base. 
Pass  two  planes,  sSs'  and  tTt',  tang,  to  the  cone  and  ||  to 
AB,   Fig.    12.     Give    Z    bet.   traces  on  drawing  for  each 
plane. 

369.  Pass  a  plane  sSs',  tang,  to  the  left  hand  cyl.  in 
Fig.  n,  and  making  an  Z   of  75°  wtih  H.     Show  element 
of  contact  and  give   Z   bet.  traces  on  drawing.     (Four  so- 
lutions.) 

370.  Pass  four  planes,  rRr',  sSs',  tTt'  and  uUu',  tang, 
to  the  left  hand  cone,  Fig.  12,  and  making  an    Z    of  6od 
with  H. 

371.  The  H  proj.  of  a  helix  with  vertical  axis  has  a 
radius  of  1.75".    Center  of  circle  is  2.25"  from  G.  L.   Curve 
rises  5"  for  each  revolution  in  a  clockwise  direction,  and 
pierces  H  plane  2.25"  from  G.  L.  on  right  hand  side  of 
axis.     The  H  proj.  of  a  pt.  R  on  the  surface  of  a  helical 
convolute  is  assumed  2"  from  the  center  of  the  H  proj. 
and  on  a  line  thru  the  center  bisecting  the  lower  right  hand 
quadrant.      Pass   a   plane,   sSs',   thru   R   and   tang,   to   the 
surface.     Give   Z  bet.  G.  L.  and  Ss'. 

372.  The  vertical  axis  of  a  helix  is  2.5"  from  V.    Gen- 
erating pt.  moves     counter-clockwise  at  a  dist.  of  2"  from 
the  axis.    Curve  springs  from  H  plane  on  left  side  of  axis. 
Pitch  of  helix  is  4".     The  H  proj.  of  a  pt.  R  on  the  sur- 
face of  a  helical  convolute  is  assumed  2.25"  from  center 
of  H  proj.  and  on  a  line  in  the  lower  left  hand  quadrant 
that  makes  an   Z    of  30°  with  the  G.  L.     Thru  this  pt.  R 
pass  a  plane  tTt',  tang,  to  the  helical  convolute.     Give   Z 
bet.  G.  L.  and  V  trace. 

373.  The  vertical  axis  of  a  helix  is  2.5"  from  V.    Gen- 
erating pt.  moves  counter-clockwise  at  a  dist.  of  2.2"  from 
the  axis.    Curve  springs  from  H  plane  on  left  side  of  axis. 
Pitch  of  helix  is  5  ".     The  H  proj.  of  a  pt.  R  on  the  sur- 


48  PROBLEMS    IN 

face  of  a  helical  convolute  is  2.5"  from  center  of  H  proj 
and  on  a  line  in  the  lower  left  hand  quadrant  that  passes 
thru  the  center  and  is  inclined  30°  to  the  G.  L.     Pass  a 
plane,  tTt',  thru  R  and  tang,  to  the  helical  convolute.    Give 
Z  bet.  G.  L.  and  V  trace. 

374.  The  H  proj.  of  a  helix  is  a  circle  of  2"  radius. 
Center  of  circle  is  2.5"  from  G.  L.    Curve  rises  3"  for  each 
revolution,   and  springs  from  H  plane  at  a  pt.  2.5"  from 
G.  L.  on  right  hand  side  of  axis.     Generating  pt.  revolves 
clockwise  as  it  rises.     The  H.  proj.  of  a  pt.  R,  on  the  sur- 
face of  a  helical  convolute,  is  assumed  2.25"  from  the  center 
of  the  circle  on  a  line  thru  the  center  in  the  lower  right 
hand  quadrant,  inclined  30°  to  the  G.  L. 

Thru  R,  pass  a  plane,  sSs',  tang,  to  the  helical  convolute. 
Give   Z   bet.  G.  L.  and  V  trace. 

375.  Pass  a  plane,  uUu',  tang,  to  the  helical  convolute 
of  Prob.  374  and  ||  to  the  line  DW,  Fig.  7.     Give  (a)  dist 
from  U  to  pt.  where  V  proj.  of  axis  intersects  G.  L.     (b) 
acute  Z  bet.  G.  L-  and  V  trace. 

376.  The  H  proj.  of  a  helix  has  a  radius  of  2.5".    Cen- 
ter of  circle  is  3"  from  G.  L.    Curve  rises  4"  for  each  revo- 
lution in  a  counter-clockwise  direction,  and  pierces  H  plane 
3"  from  G.  I/,  on  left  side  of  axis.    Assuming  this  helix  as 
the  directrix  of  a  helical  convolute,  pass  a  plane,  mMm',  |j 
to  the  line  GC,  Fig.  8,  and  tang,  to  the  helical  convolute. 
Give  (a)  dist.  from  M  to  pt.  where  V.  proj.  of  axis  inter- 
sects G.  L.     (b)  acute  Z  bet.  G.  L.  and  V  trace. 

377.  Const,  the  projs.  of  a  rt.  circular  cone  resting  on 
an  element  BE,   Fig.  8,  in  H,  vertex  at  B,  dia.  of  base 

=  I#". 

378.  Thru  the  pt.  C,  Fig.  n,  pass  a  line  tang,  to  the 
two  cyl's.     Show  the  two  pts.  of  contact,  O  and  Q.    There 
are  four  possible  solutions.     In  this  case  let  the  line  touch 
both  cyls.  on  the  side  toward  the  2nd  quadrant. 

379.  Thru  O,  Fig.  8,  pass  a  line  OF  that  shall  be  YS 


DESCRIPTIVE   GEOMETRY  49 

from  DG  and  y%"  from  AB.  There  are  four  possible  so- 
lutions. In  this  case  let  the  required  line  pass  on  the  side 
of  the  given  lines  toward  the  V  plane. 


PROBLEMS    RELATING   TO    PLANES   TANGENT  TO 
WARPED  SURFACES 

380.  Assume  the  hyperbolic  paraboloid  with  directrices 
BD  and  EM,  Fig.  8,  and  DE  and  BM  as  two  directrices. 
Pass  a  plane,  kKk',  tang,  to  the  surface  at  a  pt.  F,  on  the 
surface,  whose  V  proj.   is   %"  above   S.     Give  dist.   SK. 
Const,  double  size. 

381.  Assume  the  hyperbolic  paraboloid  of  Prob.  311. 
Pass  a  plane,  kKk',  tang,  to  the  surface  at  a  pt.  F,  on  the 
surface,  whose  H  proj.  is  i1/^'  below  S.    Give  dist.  KS. 

382.  An  hyperboloid  of  revolution  of  one  nappe  is  gen- 
erated by  revolving  DW,  Fig.  8,  about  GE  as  an  axis.   Pass 
a  plane,  kKk',  tang,  to  the  surface  at  a  pt.  F,  on  the  sur- 
face, fa"  from  the  axis  and  i"  from  V.    Give  L  bet.  traces 
on  drawing. 

383.  An   hyperboloid    of   revolution    of   one   nappe    is 
generated  by  revolving  EM,  Fig.  8,  about  ON  as  an  axis. 
Pass  a  plane,  kKk',  tang,  to  the  surface  at  a  pt.  F.  on  the 
surface,   fa"   from  the  axis  and   ^"   from  V    (above  the 
gorge   circle).     Give    L    bet.   traces  on   drawing.     Const, 
double  size. 

384.  Pass  a  plane,  tTt',  tang,   to  the  hyperboloid  of 
revolution  in  Prob.  382  at  a  pt.  J,  on  the  surface  below  the 
gorge  circle  fa"  from  the  axis  and  %"  from  V.     Give    L 
bet.  traces  on  drawing.    Const,  double  size. 

385.  Take  the  helix  of  Prob.  376  as  the  directrix  of 
a  helicoid,  whose  elements  are  inclined  45°  to  H.     Assume 
a  pt.  M  on  an  element  whose  H  proj.  bisects  the  lower  left 
hand  quadrant  of  the  circle,  and  2"  from  the  center. 

(a)   Const.  V  proj.  of  helix  for  a  half  revolution. 


50  PROBLEMS    IN 

(b)  Const,  curve  of  intersection  bet.  H  plane  and  heli- 
coid  for  a  quarter  revolution. 

(c)  Pass  a  plane,  sSs',  thru  M,  tang,  to  the  helicoid, 
giving  Z   bet.  G.  L.  and  V  trace. 

386.  Take  the  helix  of  Prob.  374  as  the  directrix  of  a 
helicoid  whose  elements  are  inclined  45°   to  H.     Assume 
a  pt.  M,  1.5"  from  the  axis,  and  on  an  element  whose  H 
proj.  is  in  the  lower  right  hand  quadrant  of  the  circle  and 
inclined  30°  to  the  G.  L. 

(a)  Const.  V  proj.  of  helix  for  a  half  revolution. 

(b)  Const.  H  trace  of  helicoid  for  a  quarter  revolution. 

(c)  Pass  a  plane,  tTt',  thru  M,  tang,  to  the  helicoid, 
giving  Z   bet.  G.  L.  and  V  trace. 

387.  Take  the  helix  of  Prob.  373  as  the  directrix  of 
a  helicoid  whose  elements  are  inclined  45°  to  H.     Assume 
a  pt.  M,  2!'  from  the  axis,  and  on  an  element  whose  H 
proj.  makes  an   Z   of  30°  to  the  G.  L.  and  is  in  the  lower 
left  hand  quadrant  of  the  circle. 

(a)  Const.  V  proj.  of  helix  for  a  half  revolution. 

(b)  Const,  curve  of  intersection  bet.  H  plane  and  heli- 
coid for  a  quarter  revolution. 

(c)  Pass  a  plane,  sSs',  thru  M  and  tang,  to  the  heli- 
coid, giving  Z  bet.  G.  L.  and  V  trace. 

388.  Assume  GE,  Fig.  8,  as  the  axis  of  the  helix  of 
Prob.  300.     Take  this  helix  as  the  directrix  of  a  helicoid 
whose  elements  are  inclined  60°  to  H.     Pass  a  plane,  kKk', 
tang,  to  the  helicoid  and  1  to  DT. 

389.  The  directrices  of  a  conoid  are :  the  circle  of  Fig. 
12,  lying  in  H,  and  the  line  BN   (produced).     The  plane 
directer  is  ||  to  the  P  plane.     Pass  a  plane,  kKk',  tang,  to 
the  conoid  at  a  pt.  on  the  surface  y2"  above  H  and  %"  from 
V.     Const,  double  size.     Give    Z    bet.  traces  on  drawing. 

390.  The  directrices  of  a  conoid  are :    the  ellipse  in 
Fig.  12,  lying  in  H,  and  the  line  DN.     The  directer  is  a 
plane  1  to  H  with  H  trace  passing  thru  Rn.    Pass  a  plane, 


DESCRIPTIVE   GEOMETRY  51 

kKk',  tang,  to  the  conoid  at  a  pt.  on  the  surface  */>"  above 
H  and  i"  from  V.    Const,  double  size. 


PROBLEMS  RELATING  TO  PLANES  TANGENT  TO 
DOUBLE  CURVED  SURFACES 

391.  A  sphere  of  3"  dia.  has  its  center  at  D,  Fig.  12. 
Pass  a  plane,  sSs',  tang,  to  the  sphere  thru  the  pt.  E  on 
the  side  nearest  V.    Give  /_  bet.  traces  on  drawing. 

392.  A  sphere  of  2"  dia.  has  its  center  at  D,  Fig.  7. 
Pass  a  plane,  kKk',  tang,  to  the  sphere  thru  the  pt.  Q  on 
the  side  nearest  V.    Give  Z  bet.  traces  on  drawing. 

393.  A  sphere  of  3^"  dia.  has  its  center  at  D,  Fig.  12. 
Pass  a  plane,  sSs',  tang,  to  the  sphere  thru  the  pt.  M  on 
the  lower  side  of  the  sphere.    Give  Z  bet.  traces  on  drawing. 

394.  A  sphere  of  2%"  dia.  has  its  center  at  C,  Fig.  n. 
Pass  a  plane,  sSs',  tang,  to  the  sphere  thru  the  pt.  B  on  the 
upper   surface   of   the   sphere.      Give   dist.    bet.    traces   on 
drawing. 

395.  A  sphere  of  2l/2"  dia.  has  its  center  in  the  G.  L. 
at  K,  Fig.   ii.     Pass  a  plane,  tTt',  tang,  to  the  sphere  at 
the  pt.  A   (A  being  on  the  surface  in  the   1st  quadrant). 
Give  dist.  KT. 

396.  A  sphere  of  3"  dia.  has  its  center  in  the  G.  L.  at 
R,  Fig.  ii.     Pass  a  plane,  tTt',  tang,  to  the  sphere  thru  the 
pt.  E  on  the  surface  (E  being  in  the  1st  quadrant).     Give 
dist.  RT. 

397.  An  ellipse  with  axes  3".  and  2"  has  its  center  at 
B,  Fig.  13,  and  its  major  axis  1  to  H.     As  it  is  revolved 
about  its  major  axis  it  generates  an  ellipsoid  of  revolution. 

Thru  the  pt.  C,  on  the  upper  part  of  the  surface,  pass 
a  plane,  tTt',  tang,  to  the  ellipsoid.    Give  dist.  a'T. 

398.  Thru  the  pt.  F,  on  the  lower  side  of  the  ellipsoid 
of  Prob.  397,  pass  a  plane,  tTt',  tang,  to  the  surface.     Give 
dist.  a'T. 


52  PROBLEMS    IN 

399.  Thru  the  pt.  E,  on  the  upper  surface  of  the  ellip- 
soid of  Prob.  397,  pass  a  plane,  tTt',  tang,  to  the  surface. 
Give  dist.  a'T. 

400.  Pass  planes,  sSs'  and  tTt',  tang,  to  the  ellipsoid 
of  Prob.  397  thru  the  two  points  on  the  surface  whose  V 
proj.  is  d' .    Give  dist.  a'S  and  a'T. 

401.  A  circle  of  2"  dia.  and  ||  to  V  has  its  center  at 
G,  Fig.  13.    As  it  is  revolved  about  AB  as  an  axis,  it  gen- 
erates a  torus. 

Thru  the  pt.  K,  on  side  furthest  from  V  plane,  pass  a 
plane,  sSs',  tang,  to  the  torus.  Give  Z  bet.  traces  on  draw- 
ing. 

402.  Thru  the  pt.  F,  on  the  upper  side  of  the  torus  of 
Prob.  401,  pass  a  plane,  sSs',  tang,  to  the  surface.     Give 
Z   bet.  traces  on  drawing. 

403.  Thru  the  pt.  C,  on  the  lower  side  of  the  torus  of 
Prob.  401,  pass  a  plane,  sSs',  tang,  to  the  surface.     Give 
Z   bet.  traces  on  drawing. 

404.  Assume  a  torus  generated  by  revolving  a  circle  of 
il/2"  dia.  about  the  line  BG,  Fig.  13,  as  an  axis;  the  center 
of  the  circle  being  1/4"  from  the  axis,  and  the  pt.  B  being- 
taken  as  the  center  of  the  torus. 

Pass  two  planes,  sSs'  and  tTt',  tang,  to  the  surface  at 
the  pts.  whose  H  proj.  is  at  c. 

405.  Assume  a  sphere  of  2"  dia.  with  center  at  B,  Fig. 
13.     Pass  two  planes,  sSs'  and  tTt',  thru  the  line  MN  and 
tang,  to  the  sphere.     Give  Zs  bet.  traces  on  drawing. 

406.  Assume  a  sphere  of  2"  dia.  with  center  at  A,  Fig. 
8.     Pass  two  planes,  sSs'  and  tTt',  thru  the  line  DG  and 
tang,  to  the  sphere.     Give    Zs  bet.  the  traces  on  drawing. 

407.  Assume  a  sphere  of  2^"  dia.,  center  in  G.  L.  at 
R,  Fig.  12.     Pass  two  planes,  sSs'  and  tTt',  thru  the  line 
OQ.  tang,  to  the  sphere.    Give  Zs  bet.  the  traces  on  draw- 
ing. 

408.  Assume  a  sphere  of  2"  dia.,  center  in  G.  L.  at  a', 


DESCRIPTIVE   GEOMETRY 


53 


54  PROBLEMS    IN 

Fig.  13.      Pass  two  planes,  sSs'  and  tlY,  thru  the  line  MN, 
tang,  to  the  sphere.     Give   Z s  bet.  traces  on  drawing. 

409.  Thru  the  line  MD,  Fig.  4,  pass  two  planes,  sSs' 
and  tTt',  tang,  to  a  sphere  of  2"  dia.  and  center  in  G.  L.  at 
G.     Give   Zs  bet.  traces  on  drawing. 

410.  Pass  two  planes,  sSs'  and  tTt',  thru  the  line  BD, 
Fig.  8,  and  at  a  dist.  of  i"  from  the  pt.  M.     Give   Zs  bet. 
traces  on  drawing. 

411.  Pass  a  plane,  sSs',  thru  the  line  MN,   Fig.    13, 
tang,  to  the  ellipsoid  of  Prob.  397  on  its  upper  side.     Show 
pt.  of  tangency,  O,  and  give   Z    bet.  traces  on  drawing. 

412.  Pass  a  plane,  sSs',  thru  the  line  MN,  Fig.  13,  and 
tang,  to  the  torus  of  Prob.  401  on  the  upper  portion  of  the 
interior  surface.     Show  pt.   of  tangency,   O,   and  give    Z 
bet.  traces  on  drawing. 

413.  Three  equal  spheres  of  dia.  iy2"  have  their  centers 
at  B,  D  and  O  respectively,  Fig.  8.     Pass  eight  planes  tang 
to  all  three  spheres.    Letter  them  from  S  to  Z  in  the  order 
of  their  distances  from  b'  measured  along  the  G.  L. 

414.  Two  equal  spheres  of  dia.   i"  have  their  centers 
at  D  and  O  respectively,  Fig.  8.     A  third  sphere  of  dia. 
il/2"  has  its  center  at  B.     Pass  eight  planes  tang,  to  all 
three   spheres.     Letter  as  in   Prob.  413. 

415.  A  sphere  of  24"  dia.  has  its  center  at  G,  Fig.  8. 
A  sphere  of  1^/2"  dia.  has  its  center  at  D,  and  one  of  i"  dia. 
has  center   at  O.     Pass    eight    planes    tang,    to  all  three 
spheres.     Letter  as  in  Prob.  413. 

416.  Const,  a  plane,  kKk',  that  shall  be  Y^'  from  B, 
Fig.  8,  y§'  from  A,  and  i"  from  D.     How  many  possible 
solutions  ? 

417.  Given  the  ellipsoid  of  Prob.  397,  const,  the  tan- 
gent cone  with  M,  Fig.   13,  as  vertex.     Show  both  projs. 
of  the  line  of  contact. 


DESCRIPTIVE  GEOMETRY  55 

INTERSECTION  BETWEEN  CURVED  SURFACES  AND 

LINES 

418.  Find  the  pts.,  O  and  Q,  where  the  line  MN,  Fig. 
n,  pierces  the  right-hand  cyl. 

419.  Find  the  pts.,  F  and  S,  where  the  line  OQ,  Fig. 
12,  pierces  the  cyl.  of  revolution  whose  axis  lies  in  BN,  and 
whose  dia.  is  i}4". 

420.  Find  the  pts.,  O  and  Q,  where  a  line   ||   to  the 
G.  L.  thru  C,  Fig.  n,  pierces  the  right  hand  cyl. 

421.  With  d  and  df  as  centers,  Fig.  n,  and  a  2"  radius, 
strike  the  lower  left  hand  quadrants  of  two  circles.     These 
are  the  H  and  V  projs.  of  a  curved  line.     Find  the  pts.,  O 
and  O,  where  it  pierces  the  left  hand  cyl. 

422.  Find  the  pts.,  F  and  S,  where  the  line  AB,  Fig. 
12,  pierces  the  cone  whose  vertex  is  Q. 

423.  Find  the  pts.,  F  and   S,  where  a  line   ||   to  the 
G.  L.,  %"  above  H  and  il/2"  in  front  of  V,  pierces  the  left 
hand  cone  in  Fig.  12. 

424.  With  q,  Fig.   12,  as  center  and   il/2"  radius,  de- 
scrib  the  upper  left  hand  quadrant  of  a  circle.     With  q'  as 
center,  and  same  radius,  describe  the  lower  left  hand  quad- 
rant of  a  circle.    These  are  the  H  and  V  projs.  of  a  curved 
line.     Determine   whether   this    curve   intersects   the   cone. 
If  so,  find  the  piercing  pts.,  F  and  S.     Draw  double  size. 

425.  Assume  a  sphere  of  2"  dia.  and  center  at  O,  Fig. 
8.     Find  the  pts.,  F  and  K,  where  the  line  GM  pierces  the 
sphere. 

426.  Assume  a  sphere  of  2l/2"  dia.  and  center  at  O, 
Fig.  8.    Assume  also  a  helix  with  axis  GE,  dia.  of  H  proj. 
2^",  and  pitch  3".     Find  all  the  pts.,  A,  B,  C,  etc.,  where 
the  helix  pierces  the  sphere. 


56  PROBLEMS    IN 

INTERSECTION  BETWEEN  CURVED  SURFACES  AND 
PLANES- DEVELOPMENT  OF  SURFACES 

427.  Assume  a  right  circular  cyl.  of  4"  dia.,  base  in 
H,  axis  _L  to  H,  and  center  of  base  2j4"  from  G.  L.     The 
cyl.  is  cut  by  a  plane,  tTt' ;  the  pt.  T  being  3"  to  the  right 
of  the  axis,  the  H  trace  inclined.  75°,  and  the  V  trace  35° 
to  the  G.  L.     (Tang.  35°=  7002.) 

Show  the  projs.  of  the  line  of  intersection.  Show  high- 
est pt.  of  curve,  E,  and  lowest  pt.,  F.  Show  pts.  of  tan- 
gency,  J  and  K,  to  left  and  right  hand  elements,  and  find 
the  true  form  of  the  curve.  Show  all  construction  lines 
necessary  to  obtain  these  various  points. 

Develop  that  part  of  the  cylindrical  surface  included 
bet.  the  H  plane  and  the  plane  tTt'.  Put  longest  element 
in  the  middle. 

428.  A  right  circular  cyl.  of  3^"  dia.  is  cut  by  a  plane, 
tTt',  inclined  60°  to  the  axis,  and  intersecting  the  plane  of 
the  base  in  a  line  2%"  from  the  axis. 

Show  the  true  size  and  shape  of  the  curve  cut  from  the 
plane,  tTt',  and  develop  the  portion  of  the  cylindrical  sur- 
face included  bet.  the  base  and  the  plane  tTt'.  Make  pat- 
tern with  longest  element  in  middle. 

429.  Make  a  pattern   for  the  section   B  in  the  stove- 
pipe elbow  shown  in  Fig.  14,  half  size. 

430.  Assume  AB,  Fig.  8,  as  the  axis  of  an  oblique  cyl. 
with  circular  base  in  H.     Dia.,  of  base   i54".     The  cyl.  is 
cut  by  the  plane  rRr'.     Show   (a)  the  projs.  of  the  curve 
of  intersection,   (b)   the  pts.  of  tangency,  J  and  K,  to  the 
outside  elements  in  the  V  proj.,  (c)  a  tang,  at  some  pt.  on 
the  curve,    (d)    the  true  shape    of    the    curve.     Show  all 
necessary  construction  lines.     Const,  triple  size. 

431.  Assume  ED,  Fig.  8,  as  the  axis  of  an  oblique  cyl. 
with  circular  base  in  H.     Dia.  of  base,  i".     The  cyl.  is  cut 
by  a  plane,  sSs'.     Show  the  projs.  of  the  curve  of  inter- 
section.    Show  also  a  right  section  made  by  a  plane  thru 
the  pt.  S  in  the  G.  L.,  and  make  a  pattern  for  that  part 


DESCRIPTIVE   GEOMETRY  57 

of  the  cylindrical  surface  lying  bet.  these  two  cutting  planes. 
Put  longest  element  in  middle  of  pattern.    Const,  triple  size. 

432.  Thru  W,  Fig.  n,  pass  a  plane,  wWw',  JL  to  the 
elements  of  the  left  hand  cyl.     Show  the  projs.  of  the  right 
section  thus  cut  from  the  cyl.     Show  the  pts.  of  tangency, 
J  and  K,  to  the  right  and  left  hand  elements.    Draw  a  tang, 
at  some  convenient  pt.  on  the  curve,  and  develop  that  por- 
tion of  the  cylindrical  surface  lying  bet.  wWw'  and  the  H 
plane.     Put  longest  element  in  middle.     Const,  triple  size. 

433.  Given  a  cyl.  with  circular  rt.  section,  radius   i" ; 
axis  ||  to  V,  2l/\."  from  V,  in  ist  quadrant,  and  inclined  60° 
to  H.     The  cyl.  is  cut  by  two  planes,  sSs'  and  tTt',  ||  to 
the  G.  L.     S^  is  3^;"  below  the  G.  L.     TY  is  4*^"  above 
the  G.  L.     The  two  planes  intersect  in  a  line  AB  in   ist 
quadrant,  2l/±"  from  H,  and  3"  from  V.     Show  projs.  of 
curve   of   intersection   bet.   planes   and   cyl.     Develop   that 
portion  of  cylindrical  surface  lying  bet.  H  plane  and  the 
cutting  planes.      Put   highest  element   in   middle. 

434.  A  rt.  circular  cone,  radius  of  base  =  2.25",  alt. 
=  4.5",  is  cut  by  a  plane  which  is  inclined  60°  to  the  axis, 
and  cuts  the  axis  in  a  pt.  2.75"  from  the  vertex,     (a)  Show 
top  view  of  curve  of  intersection,     (b)   Show  the  curve  of 
intersection  in  its  true  dimensions,  giving  lengths  of  the 
major  and  minor  axes  of  the  ellipse,     (c)  Develop  the  sur- 
face of  the  cone   with  the   curve  of  intersection ;  highest 
element  of  frustum  in  the  middle.     Give  the   Z   of  the  cir- 
cular sector. 

435.  A  rt.  circular  cone,  radius  of  base  =  2",  alt.  =  4", 
is  cut  by  a  plane  inclined  67^2°  to  the  axis,  and  cutting  the 
axis  in  a  pt.  2%"  from  the  vertex,     (a)  Show  top  view  of 
curve  of  intersection,    (b)  Show  the  curve  in  its  true  dimen- 
sions, giving  lengths  of  major  and  minor  axes  of  ellipse, 
(c)    Develop   the   surface  of  the  cone   with  the  curve   of 
intersection,  putting  highest  part  of  curve  in  the  middle. 
Give  the   Z   of  the  circular  sector. 

436.  A  cone  having  a  vertex   Z   of  90°,  and  alt.  2j4", 


58  PROBLEMS   IN 

has  a  parabola  cut  from  it  by  a  plane  whose  shortest  dist, 
from  the  vertex  of  the  cone  is  24" '.  Show  the  H  proj.  of 
the  section  and  its  true  form.  Make  a  pattern  for  that  part 
of  the  cone  belozu  the  cutting  plane. 

437.  A  cone  having  a  vertex   Z   of  60°,  and  alt.  3",  is 
cut  by  a  plane  ||  to  the  axis  and  y^'  from  it.     Show  the 
true  form  of  the  section  and  develop  the  surface  of  the 
cone,  with  the  curve  of  intersection. 

438.  Assume  A,  Fig.  8,  as  the  vertex  of  an  oblique 
cone,  and   B  the  center  of  its  circular  base  lying  in   H. 
Dia.  of  base  =  1*4' '•     The  cone  is  cut  by  the  plane  rRr'. 
Show  the  projs.  of  the  curve  of  intersection,  the  pts.   of 
tangency,  J  and  K,  to  the  V  projs.  of  the  left  and  right 
hand  elements,  and  a  tang,  to  the  curve  at  a  convenient  pt. 
Show  the  true  shape  of  the  curve.     Develop  that  part  of 
the  conical  surface  above  the  cutting  plane,  putting  longest 
element  in  middle.     Make  drawings  triple  size. 

439.  Assume  D,  Fig.  8,  as  the  vertex  of  an  oblique 
cone,  and  B  the  center  of  its  circular  base  in  H.     Dia.  of 
base  =  1^2".     The  cone  is  cut  by  the  plane  rRr'.     Show 
projs,  of  the  curve  of  intersection,  pts.  of  tangency,  J  and 
K,  to  outside  elements  in  V  proj.,  and  a  tang,  to  the  curve 
at  a  convenient  pt.      Show  the  true  shape  of  the  curve. 
Develop  that  part  of  the  conical  surface  above  the  cutting 
plane,  starting  with  the  extreme  right  hand  element.    Make 
drawings  double  size. 

440.  Find  the  curve  of  intersection  bet.  the  rt.  hand 
cone  in  Fig/ 12,  and  the  plane  tTt'.    Show  pts.  of  tangency, 
J  and  K,  with  outside  elements  in  V  proj.,  and  a  tang,  to 
the  curve  at  some  convenient  pt.     Show  the  true  shape  of 
the  curve.     Make  a  pattern  for  the  frustum  bet.  base  and 
cutting  plane,  placing  the  extreme  rt.  hand  element  in  the 
middle.     Const,  double  size. 

441.  Find  the  intersection  bet.  the  rt.  hand  cyl.,  Fig. 
ii,  and  a  prism  with  base  ned,  in  H,  edges  ||  to  V  and 
inclined  30°  to  H,  toward  the  right.     Develop  that  portion 


DESCRIPTIVE   GEOMETRY  59 

of  cylindrical  surface  lying  bet.  H,  and  a  plane   ||  to  H, 
il/2rr  above  it.     Const,  double  size. 

442.  Assume  a  rt.  handed  helix  with  axis  ON,  Fig.  8. 
Pitch   3" ;   curve   springs    from   the   pt.   E.     With  this   as 
directrix,  assume  a  helical  convolute,  and  const,  the  projs. 
of  the  curve  of  intersection   cut   by  the  plane   sSs'   from 
both  upper  and  lower  nappes. 

443.  Develop  the  surface  of  the  lower  nappe  of  the 
above  helical  convolute. 

444.  Find  the  intersection  of  the  helical  convolute  of 
Prob.  442  with  the  plane  tTt'. 

445.  Develop  the  surface  of  the  helical  convolute  of 
Prob.  444,  bet.  the  H  plane  and  the  plane  tTt'. 

446.  An  ellipsoid  of  revolution,  3"xi  y2" ,  has  its  center 
at  N,  Fig.  8;  apex  at  O;  longer  axis  1  to  H.     Find  the 
intersection  with  the  plane  tTt'.     Show  highest  and  lowest 
pts.  E  and  F.     Draw  a  tang,  at  some  convenient  pt.  and 
show  the  true  shape  of  the  curve.     Const,  double  size. 

447.  Find  the  intersection  bet.  the  plane  tTt',  Fig.  13, 
and  the  torus  of  Prob.  401.     Const,  a  tang,  at  some  con- 
venient pt.     Show  the  highest  and  lowest  pts.  E  and  F, 
and  the  true  shape  of  the  curve. 

448.  Find  the  intersection  of  the  plane,  rRr',  Fig.  13, 
with  the  torus  of  Prob.  401.     Const,  a  tang,  to  the  curve 
at  some  convenient  pt.     Show  the  highest  and  lowest  pts. 
E  and  F,  and  the  true  shape  of  the  curve. 

449.  Assume  CD  and  MN,  Fig.  8,  as  the  directrices,, 
and  the  H  plane  as  the  directer,  of  an  hyperbolic  paraboloid. 
Find  the  projs.  of  the  curve  cut  from  this  surface  by  rRr'y 
and  show  its  true  shape. 

450.  Find  the  curve  of  intersection  bet.  the  hyperbolic 
paraboloid  of  the  above  problem  and  the  V  plane  of  proj , 

451.  Assume  CD  and  MN,  Fig.  8,  as  the  directrices 
of  an  hyperbolic  paraboloid  with  the  V  plane  as  directer. 
Show  the  projs.  of  the  curve  of  intersection  bet.  this  sur- 
face and  the  plane  rRr',  and  the  true  form  of  the  curve. 


60  PROBLEMS    IN 

452.  The  directrices  of  a  conoid  are:    a  circle  in  H 
with  center  at  N,  Fig.  8,  dia.   iy2* ',  and  a  rt.  line  thru  O, 
||  to  the  G.  L.    The  plane  directer  is  the  P  plane.    Find  the 
intersection  of  the  conoid  with  the  plane  tTt',   and  show 
the  true  form  of  the  curve.     Const,  double  size. 

453.  A  shaft  running  east  and  west  on  the  second  story 
of  a  building  is  connected  by  a  belt  with  a  shaft  running 
north  and  south  on  the  first  story.     The  first  shaft  is  8  ft. 
above  the  floor,  and  the  dist.  bet.  the  shafts  is  10  ft.     The 
width  of  the  belt  is  I  ft.  and  the  dia.  of  the  pulleys  is  2  ft. 
It  is  desired  to  cut  an  opening  in  the  floor  2"  wide  of  the 
proper  form  and  in  the  proper  place  to  allow  the  belt  to 
pass  thru.     Const,  to  scale  ij^"=i'. 

454.  Assume  ON,  Fig.  8,  as  the  axis  of  a  right  handed 
helix,  dia.  of  H  proj.  =  2",  pitch  =  2^/2" ' .     It  is  the  direc- 
trix of  a  helicoid  whose  elements  are  inclined  30°   to  the 
axis.     Show  the  projs.  of  the  intersection  with  the  plane 
sSs',  const,  a  tang,  at  a  convenient  pt.  on  the  curve,  and 
show  the  true  form  of  the  curve.     Const,  double  size. 


INTERSECTION  OF   CURVED  SURFACES 

455.  The  axes  of  a  rt.  circular  cyl.  and  a  rt.  circular 
cone  are  ||  and  ij4"  apart.     Dia.  of  cyl.  =  ij4".     Alt.  of 
cyl.  =  ^A"-     Dia.  of  base  of  cone  =  4^/4" ;  alt.  of  cone 
3^/2".     Bases  of  both  figures  rest  in  H,  their  centers  being 
located  on  a  line  inclined  30°   to  the  G.  L.     Show  projs. 
of  the  line  of  intersection.     Make  pattern  for  conical  sur- 
face.    Make  pattern   for  projecting  portion  of  cyl. 

456.  A  sheet  iron  cylindrical  tank  whose  dia.  is  4  ft. 
and  alt.  4  ft.  has  a  conical  top.     Height  of  cone  =  2  ft. 
A  pipe  of  il/2  ft.  dia.  enters  from  above.     Its  axis  passes 
thru  junction  of  cone  and  cyl.  and  makes  an  Z  of  30°  with 
the  axis  of  the  cyl.     Show  projs.  of  intersection,  and  cut 
patterns  for  pipe,  top  of  tank  and  cyl.     Make  longest  ele- 
ment on  pipe  =  3  ft.     Const,  to  scale  i"=i'. 


DESCRIPTIVE   GEOMETRY  6 1 

457.  Show  projs.  of  intersection  bet.  cone  and  cyl.  in 
Fig.  15.     Const,  a  tang,  at  some  convenient  pt.  on  curve. 

458.  Find  the  intersection  of  the  cyl.  with  circular  base, 
Fig.  1 6,  and  the  cone  with  elliptical  base  (axes  2  5/16"  and 
i%")-     Draw  a  tang,  at  some  pt.  on  the  curve. 

459.  Make  a  pattern  for  spout  of  tea  pot  in  Fig.   17. 
Show  necessary  construction  work  on  paper. 

460.  Two  rt.  circular  cyls.  of  dia.  3"  have  their  axes 
y2"  apart.     They  are  each  ||  to  V,  and  the    Z    bet.  the  V 
projs.  of  their  axes  is  60°.     Find  the  curve  of  intersection 
bet.  the  two  cyls.  and  develop  the  surface  of  one  of  them. 
Place  them  in  the  most  convenient  position  with  respect  to  H. 

461.  A  vertical  tube  of  5"  outside  dia.,  3"  inside  dia., 
has  a  horizontal  cylindrical  hole  bored  thru  it  of  3"  dia. 
Dist.  bet.  axis  of  hole  and  axis  of  cyl.  =  y2".     Show  projs. 
of  outlines  of  hole  upon  a  vertical  plane  to  which  the  axis 
of  the  horizontal  hole  is  inclined  at  45°.     Cut  pattern  for 
outside  surface  of  tube,  and  show  all  necessary  construc- 
tion lines. 

462.  Let  the  elliptical  base  of  the  cone  in  Fig.  16,  be 
taken  as  the  base  of  a  cyl.  with  axis  ||  to  the  axis  of  the 
cone.     Find  the  intersection  bet.  the  two  cylinders  and  draw 
a  tang,  at  some  pt.  on  the  curve. 

463.  A  cone  of  revolution   rests   with   its  base  in  H. 
Dia.  of  base  =  4"  ;  alt.  =  4l/2r' .    Another  cone  of  revolution 
has  its  axis  inclined  45°  to  the  axis  of  the  first  cone,  inter- 
secting it  at  a  pt.  y&'  above  center  of  base  of  first  cone. 
Dist.  from  this  pt.  to  apex  of  second  cone  =  7".      Z    at 
vertex  of  second  cone  —  30°.     Show  projs.  of  curve  of 
intersection.     Develop  each  cone. 

464.  Find  the  intersection  bet.  the  two  cones  in  Fig. 
1 8,  and  const,  a  tang,  at  some  pt.  on  the  curve.     Develop 
the  surface  of  rt.  hand  cone  showing  the  curve  of  base  and 
the  two  curves  of  intersection.     Const,  to  scale   i^"— i". 

465.  Assume  a  cone  of  revolution  with  alt.  $y2"  and 
dia.  of  base  3^/2".     Its  axis  is  coincident  with  that  of  the 


62 


PROBLEMS    IN 


DESCRIPTIVE   GEOMETRY 


64  PROBLEMS    IN 

helicoid  of  Prob.  308.     Find  the  curve  of  intersection  bet. 
the  two  surfaces.     Develop  surface  of  cone. 

466.  Assume  a  cyl.  of  revolution,  dia.  il/2" ',  axis  in  Hr 
and  running  thru  the  pts.  e  and  n' ,  Fig.  8.     It  is  cut  by  the 
hyperbolic  paraboloid  whose  directrices  are  MN  and  WD 
and  whose  plane  directer  is  ||  to  H;  and  by  a  sphere  with 
dia.  2"  and  center  in  H  at  d.     Find  curves  of  intersection 
and  develop  surface  of  cyl.  bet.  hyperbolic  paraboloid  and 
sphere.     Const,  double  size. 

467.  Assume  a  cyl.  with  axis  ED,  Fig.  8,  and  circular 
base  in  H ;  dia.  of  base  2".     It  is  cut  by  a  sphere  of  3"  dia., 
center  at  N.     Find  curve  of  intersection. 

468.  The  left  hand  cone  of  Fig.  12  is  cut  by  a  sphere 
whose  center  is  at  B,  and  whose  dia.  is  2l/2f .     Show  projs. 
of  curve  of  intersection.     Const,  double  size. 

469.  The  left  hand  cone  of  Fig.  12  is  cut  by  a  sphere 
whose  center  is  at  Q  and  whose  dia.  is  3/4".    Find  curve  of 
intersection   and   develop   surface   of  cone.      Const,   double 
size. 

470.  Find  the  intersection  bet.  the  torus  of  Prob.  401 
and  an  ellipsoid  of  revolution  generated  by  an  ellipse  re- 
volving about  its   minor  axis  GM ;  the  major  axis  being 
equal  to  .3". 

471.  Find  the  intersection  bet.  the  torus  of  Prob.  401 
and  a  cone  of  revolution  with  center  of  base  at  M,  vertex 
at  B,  and  dia.  of  base  =  3". 

472.  An  ellipse   with   axes  2"  and  3"   spins  about  its 
minor  axis,  AB,  Fig.  13,  and  generates  an  ellipsoid  of  revo- 
lution.   Find  its  intersection  with  a  cone  of  revolution  with 
vertex  at  B,  center  of  base  at  M,  and  dia.  of  base  ='3". 

473.  A  torus  is  generated  by  revolving  a  circle  of  2" 
dia.  about  a  line  in  its  plane  2^/4"  from  its  center.    Let  this 
torus  rest  in  a  horizontal  plane  Y^"  above  H  and  2"  from 
V  plane,  with  axis  vertical.     Find  its  intersection  with  a 
cone  of  revolution  with  base  in  H,  axis  2"  from  axis  of 


DESCRIPTIVE   GEOMETRY  65 

torus,  dia.  of  base  =  3^>",  alt.  —  4".     Take  the  H  projs. 
of  the  axes,  on  a  line  inclined  45°  to  the  G.  L. 

474.  An   ellipsoid   of    revolution   is   generated   by   re- 
volving a  2"x3^"  ellipse  about  its  minor  axis  AB,  Fig.  13. 
Find  its  intersection  with  a  cone  of  revolution  with  base 
in  H,  axis  passing  thru  /,  dia.  of  base  =  3",  alt.  =  3^". 

475.  An  ellipsoid  of  revolution  is  generated  by  revolv- 
ing a  2"x3"  ellipse,  center  at  O,  Fig.  8,  about  its  major 
axis  1  to  H.     Find  its  intersection  with  an  oblique  cone 
with  circular  base  in  H.     Center  of  base  at  E,  dia.  of  base 
=  2",  vertex  at  M. 


MISCELLANEOUS 

476.  Thru  the  pt.  B,  Fig.  8,  pass  a  line,  BK,  that  shall 
be  i"  from  G  and  ij4"  from  D.     (Two  solutions.) 

477.  Thru  B,  Fig.  8,  pass  a  line,  BK,  that  shall  touch 
the  line  DG  and  be  il/2"  from  N.     (Two  solutions.) 

478.  Thru  D,  Fig.  8,  pass  a  line,  DK,  that  shall  be  i" 
from  O,  and  l/2ff  from  NM.     (Four  solutions.) 

479.  Assume   two   horizontal    lines,    AB    and    CD,   2" 
apart.     Their  H  projs.  make  an    L    of  60°.     With  these 
lines  as  axes,  const,  the  projs.  of  two  hyperboloids  of  revo- 
lution of  one  nappe  so  that  they  will  roll  upon  one  another 
transmitting  motion  with  a  velocity  ratio  of  2  13. 

480.  Same  as  above,  the  velocity  ratio  being  3  : 4. 

481.  A  staircase  vestibule  is   10  ft.  square.     Dist.  bet. 
first  and  second  floors  is  12  ft.     A  spiral  staircase  begins  at 
the  middle  of  one  side,  turns  thru  270°   and  ends  on  the 
second  floor.    Each  stair  is  6"  high,  the  well-hole  is  4  ft.  in 
dia.     Show  the  projs.  of  the  staircase.     Scale,  */2r'=i'. 

482.  The  pts.  O,  B  and  D,  Fig.  8,  are  the  terminations 
of  three  concurrent  edges  of  a  rectangular  parallelepiped. 
Const,  its  projs. 


DESCRIPTIVE   GEOMETRY  67 


ANSWERS  AND  OTHER  DATA 

Problems  in  descriptive  geometry,  when  worked  by  stu- 
dents on  plates  of  uniform  size  should  be  so  located  that 
they  will  not  interfere  with  one  another,  nor  extend  beyond 
the  limits  of  the  sheet.  To  facilitate  the  proper  placing  of 
the  problems,  we  give  here,  in  addition  to  the  answer  for 
each  problem,  the  size  of  the  rectangle  in  which  the  prob- 
lem may  be  worked,  and  the  coordinates  of  some  known 
point  within  the  rectangle. 

To  illustrate  the  use  of  these  data,  let  us  suppose  that 
the  paper  on  which  the  problems  are  to  be  worked  is  of 
the  size  I2"xi6",  and  that  it  is  desired  to  have  the  student 
solve  Problems  46,  55,  62  and  69  on  that  sheet.  Refering 
to  Prob.  46  in  the  following  pages,  it  is  seen  that  it  may 
be  worked  within  a  space  6"x4*/2"  (the  first  dimension  being 
the  horizontal  one),  the  point  G  being  located  2^6"  from 
the  left  edge  and  2%"  from  the  lower  edge.  Now  let  a 
6"x4J/>"  rectangle  be  cut  from  a  sheet  of  paper  and  the 
point  G  located  thereon  by  means  of  its  coordinates.  Sim- 
ilarly for  Problems  55,  62  and  69.  These  pieces  of  paper 
may  then  be  placed  upon  the  I2"xi6"  sheet  in  the  most 
convenient  positions,  and  the  coordinates  of  the  points  G, 
S,  etc.,  measured,  taking  the  lower  left  hand  corner  of 
the  plate  as  origin.  The  specifications  may  then  be  placed 
upon  the  blackboard  in  the  following  brief  manner: 

Plate  4 

Prob.  46.     Put  G  at  (3^",  9y4"). 
Prob.  55.     Put  G  at  (8^",  9%"). 
Prob.  62.     Put  ri  at  (i",  2#"). 
Prob.  69.     Put  S  at  (12^",  4"). 
If  the  answer  to  a  problem  is  a  linear  measurement  and 


68  PROBLEMS    IN 

it    has    been    constructed    "double   size,"    the    measurement 

should  be  divided  by  two. 

Problem. 

13.— 2^x4%;  K  at  (ift,  3%). 

14.— 4x554;  G  at  (#,  25/6). 

I5-— 3^x5^;  d  at  (ft,  %). 

16.— 5^x5^;  S  at  (3%, 

17.— 2^x3^4;  R  at  (#, 

18.— 4^x4Ks;  K  at   (3,   i 

I9._ 5^x5^;  S  at  (3#, 

20.— 2^x3^;  R  at  (i%, 

2i.—iy2xi%;  S  at  (194 

22.— 1^x1%;  S  at  (ift, 

23.— i J4x2^;  K  at  (^,  ft). 

24.— i^xiK;  G  at  (J^,  5^). 

25-— 3^x4^;  G  at  04  2).     Ans.  2.80". 

26.— 2^x5;  G  at   (%,  2ft),     Ans.  1.87". 

27.— 3^x3^;  K  at  (%,  i%).     Ans.  2.51". 

28.— 3^x3^;  S  at  (i,  i%).    Ans.  3.37". 

29.— 43/6x3%;  R  at  (J4,  2).     Ans.  3.76r/. 

30.— 4^x4%;  S  at  (IJ4,  tf/s).     Ans.  1.03". 

31—3^x3^;  G  at   (ft,  i#).     Ans.   1.93". 

32.— 2^x3 >4  ;  K  at  (%,  2).     Ans.  1.97". 

33-— 3^x3%;  S  at  (i#,  2J4).     Ans.  0.94". 

34.— 2^x5;   K   at    (1%,   32/4).     Ans.    u°2g'   with   H, 

63°35'  with  v- 

35.— 4j4x4j^  ;  G  at    (J^,   2^).     Ans.   2o°s8'   with  H, 

I5°34'  with  V. 

36.— 4^4x3^4  ;    S   at    (i,    i%).     Ans.    2i°45r   with   H; 
54°37'  with  V. 

37.— i#x2j4;  d  at  (^,  ^).     Ans.  0.82". 

38.— 3J4MH;  R  at  (J^'  J^)-     Ans-  a88"' 
39.— 434x4^4  ;  d  at  (^,  >i).     Ans.  o.8ir/. 
40.— 4^4x4^4  ;  d  at  (J^,  ft).    Ans.  o.6ir/. 
4I._3i4xs;  G  at   (ft,  3%).     Ans.  6i°35'  or  its  sup- 
plement. 


DESCRIPTIVE    GEOMETRY  69 

42.— 3%x4%;  R  at  (J/s,  2^).     Ans.  51°  or  its  sup. 
43.— 4x4%;  S  at  (3%,  3^),     Ans.  80° 5'  or  sup. 
44. — 6%x6% ;  G  at  (2%,  i%).    Ans.  I9°3o'  or  sup. 
45-—3^x4%;  G  at  (ift,  i^).    Ans.  o°. 
46. — 6x4%;  G  at  (2^,  2%).     Ans.  59°io'  or  sup. 
47.— 4x4^;  S  at  (3%,  314).     Ans.  5i°o'  or  sup. 
48.— 4^x3%  ;  K  at  (i/,  i%).    Ans.  73°44'  or  sup. 
49-— 5^x3%;  R  at  (ij4,  2%).    Ans.  83°2o'  or  sup. 
50.— 2%x5%  ;  K  at  (\Y&,  3%).     Ans.  39°45'  or  sup. 
51.— 4^x3;  S  at  (2%,  \y&).    Ans.  3°45'  or  sup. 
52.— 3%x4%;  G  at  (#,  i#).     Ans.  3.37". 
53.— 4x2^;  R  at  (*/$,  i^).     Ans.  0.15". 
54.— 2^x4^;  G  at  (#,  3%).     Ans.  56°. 
55-— 6^x3:  G  at  (Y&,  i3/4).    Ans.  78°3o'. 
56.— 594xs%;  S  at  (2%,  i%).     Ans.  88°4o'. 
57.— 6x654  ;  S  at  (2#,  3^).    Ans.  51  "25'. 
58.— 314x6%  ;  G  at  (J^,  2%).     Ans.  36=42'. 
59.— 4%x6;  R  at  (#,  3#).    Ans.  i32°53'. 
60.— 4x2%;  R  at  (ys,  i5/8).    Ans.  79°  10'. 
61.— 4X5J4;  G  at  (J4,  2%).    Ans.  74°3o'. 
62.— 554x5%;  »t'  at  (Y&,  2l/s).    Ans.  29°2o'. 
.^3--3x3K;  R  at  (%,  2%).    Ans.  520i5'. 
64.— 254x5;  U  at   (ys,  i3/&}.     Ans.  64°2i'. 
65.— 2%x6;  G  at  (i%,  3).     Ans.  82°34'. 
66.— 4^x3%;  K  at  (ij£,  i%). 
67.— 5^x5%;  S  at  (2S/S,  i.%). 

68.— 8-^x5^;  S  at   (*/2,  3^).     Ans.  o  =  1.95";  g  -- 
2.30"  ;  d  ==  1.44". 

69.— 8%x5%;  S  at    (5>/2,  3).     Ans.  a  =   1.95";  d  = 

3-35":  .?  ==  2.14". 

70.— 5x3%;  R  at   (ij4  2%).  Ans.   ii°io'. 

71.— 5%x4/2;  G  at  (J$,  3%).  Ans.  io°io'. 

72.— 734x6%  ;  R  at  (Y&,  3%).  Ans.  49°4o'. 

73.— 6x6/2;  S  at  (2ys,  3%).  Ans.  53°45'. 

74.— 3Mx6%;  G  at  (#,  2%).  Ans.  i6°is'. 
75.— 4%x6;  R  at  (Y&,  3^)-    Ans.  64° 30'. 


70  PROBLEMS    IN 

76.— 2x2^;  G  at  (#,  i%).     Ans.  32°  15'. 

77.— 2^x4^;  G  at  (#,  i#).    Ans.  33°2o'. 

78.— 5%x4#  ;  G  at  (2^,  2%).    Ans.  460is'  or  149°  15'. 

79.— i%x3#;  G  at   (#,  i#).     Ans.  I3O°45'- 

80.— 7x754  ;;K  *t  (i&  3Mi).    Ans.  o.io". 

81.— 3^x4%;  R  at  (ys,  21/4).     Ans.  1.26". 

82.— 5^x6%;  G  at  (ys,  3#).     Ans.  1.05". 

83.— 4x2%  ;  R  at  (2^,  ij£).    Ans.  0.19". 

84.— 2/xi^;  U  at  (#,  i%).     Ans.  1.31". 

85.— 4^x2/;  K  at  (3#,  ^).     Ans.  1.69". 

86.— 2^x3^  ;  d  at  (%,%).     Ans.  1.02". 

87.— 1^x3;  W  at  (i^,  i%). 

88.— 7x5^  ;  K  at  (ij4,  2).    Ans.  i.oo". 

89.— 3^x4^;  R  at  (#,  2^).     Ans.  2.19". 

9o._6i4x5^  ;  R  at  (5^,  3%).     Ans.  0.23". 

91.— 6x6^;  G  at  (J&  3^).     Ans.  0.62". 

92.— 4^x4^  ;  R  at  (%,  1 5^).    Ans.  0.49". 

93. — 1^x3;  R  at  (lj4,  a).     Ans.  1.89"  below  G.  L. 

94.— 2x3^;  q'  at   (i,  i#).     Ans.  0.15". 

95.— 2x1 1  ;  R  at  (i%, 

96.— i%x3%;  (/'  at  (34, 

97.— 2^x2%;  R  at  (i^ 

98.— 5x4;  Gat  (^,  2^). 

99.— 7x8/2;  S  at  (454,  2 J4). 

loo.— 3^4x4%  ;  U  at  (J4,  2#). 
ioi.— 3x3^  ;  R  at   (i5/£,  i^).     Ans.  0.65". 
102.— 6x5%;  S  at  (3%,  3]&),     Ans.  0.24". 
103.— 3^4x4^4  ;  K  at  (2%,  2§/£)-    Ans-  I-52"- 
104.— 2^x3;  U  at  (ys,  i^).     Ans.  0.62". 
105.— 3x5;  S  at  (2ys,  ift).    Ans.  1.05". 
106.— 3^x3^4;  U  at   (2j4,   ij4).     Ans.   1.96". 
107.— 3x5^  ;  d  at  .( J4,  y4).    Ans.  2.06". 
108.— 2/X2J4;  M  at  (ij4,  i).     Ans.  0.56". 
109.— 1^x3;  d  at  (^,  H)-     Ans-  i-37w- 
no.— ixi/  ;  W  at  (^4,  %).    Ans.  0.5". 
in.—  Ans.  1.73". 


DESCRIPTIVE   GEOMETRY  71 


H2. —  Ans.  1.15". 

113. — 4x6;  U  at  (l/s,  2/4)- 

114.— 55^x4;  R  at  (i,  2J4). 

115.— 2^x3 24  ;  d  at  (5^,  i^g). 

1 16.— 5^x45^;  W  at  (3,  2J4). 

117.— 4^4x9%;  S  at  (i^,  5J4).    Ans.  39°4o'. 

"8.— 7#xn#;  S  at  (2^,  534).     Ans.  158°  15'. 

119.— 4x3%;  K  at  (54  2).     Ans.  0.45". 

120. — 4x4%;  K  at  (il/2y  2%).    Ans.  1.28". 

i2i.— 4^4x3%;  K  at  (J/g,  i^).     Ans.  4.49". 

122.— 35/8x344;  G  at  (%,  i^4).     Ans.  0.17". 

123.— 2%x4%  ;  S  at  (5/,,  2%).     Ans.  1.5". 

124.— 334x3%  ;  G  at  (%,  L%).    Ans.  3". 

125.— 3x5%  ;  c'  at  (o,  3^4).     Ans.  4.06". 

126. — 1^x2^;  R  at  (2,  2^6).     Ans.  0.56". 

127. — 4x5  ;  K  at  (ij^,  2^).    Ans.  Maj.  2.25"  ;  Min.  0.41' 

128. — 7^x4;  G  at  (J^,  2%).    Ans.  7.6o/r. 

129.— 6^x5%;  T  at  (3%,  4).     Ans.  0.98". 

130.— 3%x654;  S  at  (/,  4^).     Ans.  3.14". 

131.— 4^x4%;  G  at  (^,  3).    Ans.  2.11". 

132-— 3J4x4}4  ;  R  at  (5^,  3^).    Ans.  1.5". 

133.— 4x3^4  ;  K  at  (ij4,  2).     Ans.  0.43". 

134.— 4x754  ;  S  at  (23/8,  5^).    Ans.  0.33". 

136-— 3J4X3J4  ;  U  at  (j^,  154).     Ans.  0.99". 
137. — 4x4;  K  at  (5^,  i^i).    Ans.  1.38". 
138. — 6x5;  R  at  (3,  2^5).    Ans.  2.83". 
I39-— 6^x5;  U  at  (354,  2%).     Ans.  3.12". 
140.— 5/X4/;  K  at  (i%,  1^4).     Ans.  1.54". 
141.— 6^x5%  ;  K  at  (4^,  4^).     Ans.  0.17". 
142.— 10x8%;  S  at  (554,  6%).     Ans.  3.39". 
143.— 2^x4^4;  K  at  (i^,  3%).     Ans.  2.92". 
144.— 45^x5;  G  at  (5^,  2^).     Ans.  0.14". 
145.— 4x7;  K  at  (2%,  i%).    Ans.  0.71". 
146.— 454x8;  K  at  (i#,  3J4).     Ans.  0.56". 
I47-— 3^4x524 ;  K  at  (i#,  3^),    Ans.  0.91". 


72  PROBLEMS    IN 

148.— 7x6j4;  S  at  (3^,  3).     Ans.  2.03". 
149.— 5^x6;  V  at  (3%,  3^).    Ans.  2.01". 
150.— IJ4X224;  K  at  (}4  l>£).     Ans.  0.96". 
151.— 4>4x6>4;  G  at  (i^,  3%).     Ans.  2.60". 
152.— I24X3J4;  G  at  (*/g,  i^).     Ans.  1.12". 
I53-—  i#x  2%;  G  at  (#,  i%).     Ans.  1.80". 
154.— 4x5;  S  at  (i,  ii/2).    Ans.  0.99". 
155.— 4x4^;  U  at  (ft,  i#).     Ans.  43°io/. 
156-— 5#*6}4;  S  at  (H>  4J4)-    Ans.  74°io/. 
I57-— 5^x6;  <*  at  (24,  24)-     Ans.  12°. 
158.— 4x3;  W  at  (i^j,  i%).     Ans.  38°4o'. 
I59-— 4%xs;  R  at  (2y2,  2J4).     Ans.  26°34/. 
160.— 554x2%;  R  at  (3%,  i%).     Ans.  63°2o'. 
161.— 5x3^;  Gat  (i^,  i%). 
162.— 2^x3%;  R  at  (#,  i). 
163.— 3^x3%;  G  at  (%,  2J4). 
164.— 4x4;  R  at   (}£,   ij^).       * 
165.— 5x5;  S  at  (2,  2^). 

1 66. — 2^x2%;  Left  end  of  G.  L.  at   (J^,   ift).     Ans. 
0.60". 

167. — 434x4%  ;  Intersec.  bet.  G.  L.  and  profile  plane  at 

2%).     Ans.   1.45". 
1 68.— 424x4;  R  at  (i%,  2j4). 
169.— 5x7;  R  at  (il/2,  2). 
170.— 2x4%;  U  at  (ft,  i%). 
171.— 3x4%;  U  at  (ft,  i%). 
i?2.— 5^x3/2;  G  at  (i,  i%). 
I73-— 3/2x5M;  S  at  (J4,  3). 

174.— 5^x  3J4;  G  at  (i,  i%).     Ans.  7O°45'  or  sup. 
175. — 2%x3%  ;  R  at  (ft,  i).     Ans.  8o°io'  or  sup. 
176-— 3/4x3^;  S  at  (ft,  2^).     Ans.  75°i5r  or  sup. 
177. — 2%x2%  ;  G.  L.  i^"  up.     Ans.  57°  or  sup. 
178. — 4^x4%  ;  Intersec.  bet.  G.  L.  and  profile  plane  at 

ft).    Ans.  32°5or  or  sup. 

179. — 6x2;  R  at  (5%,  1 2^).    Ans.  46°  15'  or  sup. 
1 80. — 5x2%  ;  U  at  (3%,  i).    Ans.  6i°2o'  or  sup. 


DF^CRIPTIVE  GEOMETRY  73 

181.— 4x6;  R  at  (%,  4%).     Ans.  39°  14'  with  H,  63°26' 
with  V. 

182.— 354x454;  U  at   (56,   i J4).     Ans.  73°54/  with  H, 
33°4i'  with  V. 

183.— 3^x354  ;  K  at   (i%,  34).     Ans.  49°45'  with  H, 
67°45'  with  V. 

184. — 6xio;  K  at  (i,  5).     Ans.  6s°45'. 

185.— 5x6;  U  at  (4%,  2  J4).     Ans.  50° 46'. 

186.— 4M*3J4;  G  at  (5$,  i%).     Ans.  5°4o'. 

187.— 5x454  ;  S  at  (54,  25*).    Ans.  45°2O/. 

1 88.— 3x5;  Sat  (15^,254). 

189.— 3x254  ;  G  at  (56,  i%).    Ans.  1.30"  below  G.  L. 

190. — 3^x256  ;  S  at  (i%,  2).     Ans.  1.95"  below  G.  L. 

191.— 454x5%;  S  at  (%,  2%). 

192.— 2^x3^6;  G  at  (%,  2%). 

J93-— 7H*7s/s  ;  U  at  (2%,  3%).    Ans.  5i°45'  or  6i°2o'. 

194.— 6%x8^  ;  S  at  (3%,  4).    Ans.  I9°3o'  or  3O°5o'. 

I95-— 6>4x554  ;  K  at  (1-56,  3%).     Ans.  -0.8". 

196.— 954x1054  ;  S  at  (2%,  4%).     Ans.  0.88". 

197.— 654x954  ;  S  at   (2%,  5).     Ans.  0.46". 

198.— 6x414  ;  G  at  (2%,  2%).     Ans.  0.60". 

199.— 2^4x2%;  G  at  (%,  i%).    Ans.  2.29". 

200.— 4^x8;  G  at  (%,  2>4).     Ans.  1.95". 

201.— 354x254;  d  at  (2%,  %). 

202.— 4^4x554 ;  R  at  (254, 

203.— 654x5^ ;  R  at  (254, 

204.— 654x5^;  R  at   (2y4,  3%). 

205.— 7x5%  ;  &'  at  (454,  3#)- 

206.— 8%x654;  S  at  (554, 

207.— 2^x5;  K  at  (5^,  2). 

208.— 454x3%;  n  at  (2%,  i%). 

209.— 45^x5 54  ;  w  at'  (%,  i%). 

210. — 734x7%  ;  G.  L.  4"  up.    Ans.  0.577  ft. 

211. — 724X7>4 ;  G.  L.  4"  up.    Ans.  0.27". 

212. — 7%x754  ;  G.  L.  4"  up.    Ans.  0.92". 

3X-x3^  ;  T  at  (>6,  ij4).    Ans.  3304i'. 


74  PROBLEMS    IN 

214.— 3x4^4  ;  T  at  (ys,  iy2).    Ans.  0.707". 
215.—  Ans.  58°49'. 

216.—  Ans.  47°2/. 

217-— 5x554 ;  S  at 

218.— 2x234;  K  at  ( 

219.— 234x3}^;  R  at 

220.— 5x5^;  rf  at  (z%,  2%).     Ans.  55°. 

221.— 714x8^;  V  at  (54,  334). 

222.— 4^x4^  ;  S  at  (134,  2i/g). 

223.— 234x3;  G  at  (/,  i#). 

224. — 6x6/  ;  S  at  (2,  1 34).    Ans.  1.21". 

225.-8/>x7;  U  at  (5^,  6). 

226.— 8>ix7^;  R  at  (3/4,  334). 

227.— 6j4x8>4  ;  R  at  (4Y&, 

228.— 534x11/2;  U  at  (ys, 

229.— 414x414  ;  V  at  (i%,  2.%-). 

230.— 4^x4.%  ;  a  at  (%,  iy4). 

231.— 6y4x43/4  ;  V  at  (iy&,  2$/8). 

232.— 71/4x11;  V  at  (2%,  6J4). 

233-— 3^x5;  S  at  (i,  2^). 

234.— 414x8%  ;  Middle  of  H  proj.  at 

235-— 5^x7/2 ;  R  at  (j^,  i%). 

236.— 4^x554  ;  G  at 

237-— 3>4x4>4  ;  G  at 

238.— 4>4x6%  ;  S  at 

239-— ?K'X9^:  c  at  (i>4,  334). 

240.— 3x4. 

241.— 5x534. 

242.— 4x5. 

243.— 9/2x7-%  ;  U  at  (234,  3%). 

244.— 8/xii3^;  R  at  (i/,  6). 

245.— 9x11%;  R  at  (\y2,  6). 

246.— 6^x9;  c'  at  (25/s, 

247-— 5^x8%  ;  c'  at  (i 

248.— 6%x6/  ;  R  at  (2%,  234), 

249.— 4x23/6;  R  at  (i^,  i/). 


DESCRIPTIVE   GEOMETRY  75 


250-—  sY^A  ;  G  at 

251.—  16x12;  G  at 
252.—  7x524;  G  at  (Y&, 
253-—  6^x65/6  ;  c'  at 
254.—  9x8^4  ;  c'  at  (4^, 

255-—  4^*7^4;  &'  at  (24,  424). 

256.—  4^x5;  W  at  (2,  3>4).     Ans.  0.84". 
257-—  7^x9^;  V  at  (#,  53/4).     Ans.  1.66". 
258.—  3#x4K;  &'  at  0-6,  2^).    Ans.  1.49". 
259.—  424x8  ;  b  at  (%,  2%).    Put  fcc  ||  to  G.  L.  and  2j^" 
below  it.    Ans.  2.04". 

260.  —  424x8;  be  as  in  Prob.  259.    Ans.  1.63". 

261.—  4X6J4:  6'  at    (Y&,  4^).     Ans.    (a),  0.09";    (b), 

1-95". 

262.—  5^x7^;  <?'  at  (ij4,  3J4).  Ans.  (a),  0,75";  (b), 
0.99". 

263.—  6x6l/2  ;  b'  at  (i^$,  3).  Ans.  Rad.  =  1.98".  Center 
1.62"  above  H. 

264.  —  9^x1054  ;  b'  at  (2,  6y2).     Ans.  0.45". 

265.—  8^x12%  ;  6'  at  (i,  8^).  Put  be  \\  to  G.  L.  and  2" 
below  it.  Ans.  0.51". 

266.—  A  =  59°4';  B  =  88°  12';  a  =  so°2'. 

267.—  B  =  90°  ;  C  =  45°44'  ;  c  =  45°  12'. 

268.—  A  =  8o°2o';  B  =  73°47';  b  =  6i°47'. 

269.—  A  =  5i°58';  C  =  83°55';  c  —  51°. 

270.—  a  =  68°46';  c  =  63°  12';  A  =  74°. 

271.—  b  =  63°  15';  c  =  47°42';  C  =  55°53'. 

272.—  a  =  38°  ;  b  =  42°  ;  B  =  58°53'. 

273.—  b  =  48°3'  ;  c  =  64*47'  ;  C  =  8o°2o'. 

274.—  A  ==  75°54';  B  ==  49°i6';  c  =  63°36'. 

275.—  A  =  82°9';  B  =  90°  ;  c  =  45°  12'. 

276.-B  =  88°  12'  ;  C  =  55^3'  ;  a  =  5o°2'. 

277.—  A  =  8o°2o'  ;  B  =  73°47'  ;  c  =  48°3'. 

278.—  A  =  82°9';  b  =  82°i7';  c  =  45°  12'. 

279.—  b  =  63°  15'  ;  c  =  47°  42'  ;  A  =  59°4'. 

280.—  a  =  51°  ;  b  =  38°  ;  C  =  58053'. 


76  PROBLEMS    IN 

281.—  b  =  61°  47'  ;  c  =  48°3'  ;  A  =  8o°2o'. 

282.—  A  =  83°55';  B  =  5i058';  C  =  58°53'. 

283.—  A  =  67°  ;  B  =  74°  ;  C  =  82°. 

284.—  A  =  55°53';  B  =  59V;  C  =  88°i2'. 

285.—  A  =  54°8';  B  =  8o°2o';  C  ==  73°47'. 

286.—  a  =5i°;b  =  T,8°;c  =  42°. 

287.—  a  =  48°3'  ;  b  =  64°  47'  ;  c  =  61°  47'. 

288.—  a  =  45°  12';  b  =  79°i';c  =  82°  17'. 

289.—  a  =  5o°2';  b  =  63°  15';  c  =  47°42'. 

290.—  9^x7;  p  at  (2#,  #).  Take  pp'  =  4^";  >#" 
—  5*4".  Ans.  A'  =  3304i';  B'  =  36^2';  C'  =  2i°48'; 
D'  =  47°58';  E'  =  48°  i';  F'  =  26°2i';  G'  =  36°29'  ; 
H'  =--  4i°44'  ;  I'  =  44°5o'  ;  ]'  =  -  i8°8'  :  K'  =  7i°3o'. 

291.—  8y2x6y4;  I  at  (4^,  2^).  Ans.  A'  =  33°4i'; 
B'  =  56°  19';  C  =  29°  i';  D'  =  29°  i';  E'  ==  38°42';  F'  = 
17°  55';  G'  =  20°  18';  H'  =  27°29';  I'  =  27=50';  ]'  = 


292.—  6x6%;  /  at  (2%,  2>^).     Ans.  A"  =  45°;  B" 
33°4i';  C"  =  29°!';  D"  =  46*41';  E"  =  6o059';  F"  = 
36°2';    G"  =  46°4i';    H"  =  36°2';    I"  =  4o037';    }" 
-9°30';  K"  =  6403o'. 

293.  —  14x12;  w  at  (4J/i,  4l/z).    Take  \m'  =  4".    Ans. 
A"  =,  45°  ;  B"  =  6703o'  ;  C"  =  42°44'  ;  D"  =  i6°5'  ;  E"  = 

4705';F"  =  i5035';G"  =  i605';H"=i5°33';I"=i5035'; 
]"  =  —  1°5;;   K"  =  86°. 

294.  —  12x16;  «'  at  (2ys,  8).       Take  ym'  =  3".       Ans 
A"  =  33°4i';  B"  =  69°  10';  C"  =  32°;  D"  =  17^5'  ; 
E"     -  35°3o';  F"  =  n°3o';  G"  =  12°;  H"  =  i7°io': 
I"  ==  17°  n';  J"  =  --  i°s';  K"  =  93°3o'. 

295.—  111^x814  ;  w  at  (4^4,  y8).  Take  ym'  =  3".  Ans. 
A'"  =  38°40';  B"'  =  50°5o';  €'"  =  32°;  D"'  =  32°i5'; 
E'"  =  45°35'  ;  F'"  =  23°  10'  ;  G'"  =  26°5o'  ;  H'"  =  29°  10'  ; 
I'"  =  30°;  J'"  =  -6°5o';  K'"  =  78°. 

296.—  12x16;  p  at   (Y2,  6l/2).     Take  ym'  =  4".     Ans. 
A  =  38°40';  B  =  5i°2o';C  =  32°;  D  =  32° 


DESCRIPTIVE   GEOMETRY  77 

F==22°s8';  G  =  68°ii';  H  =  29°i2';  I  =  29°54'; 
J=— 48°25';  K  =  78°. 

297. — 16x12;  p  at  (3%,  l/2).  Take  vm'  =  4".  Ans. 
A'  =  45° ;  B'  =  38°4o' ;  C  =  32° ;  D'  =  4i°28' ;  E'  -  58°  ; 
F'  =  33°3i':  G'  ==  4i°28';  H'  •*  3303i';  I'  =  36°23'; 
J'=  -i5°5';  K'  =  64°4o'. 

298. — 12x12;  /  at  (3^,  234).  Take  nm"  =  3".  Ans. 
A'  =  26°34' ;  B'  =  58° ;  C'  =  22°59' ;  D'  =  29°  12' ;  E'  = 
3o°3i';  F'  =  i3043';  G'  =  i5°37';  H'  =  28°  18';  I'  = 
28°3o';  J'  =  -3036';K'  =  82°45'. 

299. — 16x12;  /  at  (10,  3J4).  Take  nm"  =  3".  Ans. 
A"  =  38°40';  B"  =  32° ;  C"  =  22°59';  D"  =  5i°2o';  E" 
.56°29';  F"^3i°59';  G"  =  44°59';  H"  =  4i°28'; 
I"  =  46°39' ;  J"  =  — 23°4o' ;  K"  =  6S04o'. 

300.— 3.^x7^  ;  Center  of  H  proj.  at  (3^,  3^). 

301. — 6xioj4  ;  Center  of  base  of  upright  cone  at  (3,  3). 

302.— 214x514 ;  Center  of  H  proj.  at  (i^,  iy&). 

304.— 9^x7^;  e'  at  (4%,  3^). 

305.— 454x3^;  Center  of  base  at  (2j4,  ^). 

306. — 5x4^  ;  Center  of  H  proj.  of  base  at  (2l/2,  fa). 

307.— 3x3';  R  at  (2J4,  iK).     Ans.  4i°48'. 

308. — 4x6)4 ;  Center  of  H  proj.  at  (2%,  i%). 

309.— 7x5^/4  ;  R  at  (IJ4,  2ji). 

310.— 234x414;  7£/  at  (i^, 

311.— 6^x4%;  ^  at  (IJ4, 
312.— 454x414  ;  S  at  (2%,  2y&).    Ans.  0.95". 
313.— 6^x7;  Rat  (2,  3%). 
314.— 2^4x514  ;  w'  at   (i^, 
315.— 8x7;  S  at  (sl/2,  5). 
316.— 834x754  ;  &'  at  (154, 
317.— 134x414  ;  n'  at  (%, 
318.— 134x434;  „'  at  (%, 
319. — 4x1034;  Center  of  H  proj.  at  (2%,  434). 
320. — 434x934  ;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 
at  (2%,  654). 

321.— 354x334;  K  at  (134,  2).    Ans.  30°  10'. 


7§  PROBLEMS    IN 

322.— 3%x42i;  K  at  (i%,  2%).     Ans.  55°3o'. 
323.— 4j4xsH;  R  at  (2%,  2%).     Ans.  36°2o'. 
324.— 6^x4;  R  at  (4^,  2%).     Ans.  30°  10'. 
325.— 7x4^4  ;  K  at  (3%,  2^).     Ans.  60°. 
326.— 324x3%;  c  at  (2,  %).    Ans.  1.48"  and  0.85". 
327.— 4x4;  e'  at  (2,  224).    Ans.,  H  trace,  0.96",  V  trace, 
0.72". 

328.— 5x5;  K  at  (2%,  224).    Ans.  147°  16'  or  sup. 

329.— 4%xs;  K  at  (%>  3)-    Ans.  52°25'  or  sup. 

330.— 5x8;  K  at  (3%,  5%).    Ans.  78°  and  67°45'- 

331-— 424x4^8  ;  R  at  (2^4,  2%).    Ans.  42°  10'  and  49°5o'- 

332-— 724x524  ;  R  at  (3%,  2%).     Ans.  i8°3o'. 

333.— 5/2x3%  ;  K  at  (5^,  i%).    Ans.  5° 5'. 

334.— 6x5%;  q  at  (3^;  i%).     Ans.  2.02". 

335-— 4^x4;  a  at  (2^,  J^).    Ans.  0.35". 

336.— 4^x4;  W  at  (4J4,  2%).    Ans.  0.05". 

337.— 4^x3%;  R  at  (2^,  2%).     Ans.  0.98"  and  1.47". 

338-— 5#x4#;  K  at  (2J4,  2%).     Ans.  46°  10'. 

339-— 4^2x5^;  K  at  (ij4,  3).    Ans.  i8°3o'  or  sup. 

340.— 5^4*624  ;  R  at  (3,  2%).    Ans.  15° 45'  or  sup. 

341. — 5x4;  K  at  (%,  2).     Ans.  2°io'. 

342.— 5x4;  K  at  (%,  2).     Ans.  i4°55'- 

343-— 6#X4J4;  K  at  (2^,  2><).     Ans.  i4°so'. 

344.— 3x5^  ;  R  at  (2^,  2/8).    Ans.  3.46"  or  1.30". 

345-— S^M^  ;  o  at  (J^,  ^).    Ans.  0.32"  or  1.08". 

346. — 434x4>4;  o  at  (34 »  I/7/s)-     Ans.  0.32" 'or  0.76". 

347-— 5/4x5/4  ;  o  at  (2%,  i^)-     Ans.  o.io"  or  2.18". 

348.— 9x7^;  R  at  (5^,  4%).     Ans.  2O°45'  or  sup. 

349.— 5x8%  ;  K  at  (i%,  4^)-     Ans.  2°i5'. 

350.— 8x4^;  R  at  (4^4,  2%).     Ans.  i403o'. 

351.— 724x7%;  R  at  (35/s,  4ys).    Ans.  5°. 

352.— 10^x6^  ;  b  at  (2%,  i^).     Ans.  20° 50'. 

353.— 11x5^;  W  at   (4%,  2%).     Ans.   I2°so'. 

354.— 22^x3%  ;  W  at  (2$*,  124).     Ans.  66°. 

355-— 324x4%:  K  at  (2,  2%).     Ans.  33°45'  and  77°2o'. 

356.— 9x4%  ;  R  at  (6%,  2#j).     Ans.  i8°3o'. 


DESCRIPTIVE  GEOMETRY  79 

357-— 8j4xs$4  ;  K  at  (3^,  3%).     Ans.  32°2o'. 
358.— 324*7%;  R  at  (2%,  2^).     Ans.  39°4o'- 
359.— 10^x6^ ;  W  at  (2%,  3%).    Ans.  14°  and  21°. 
360.— 13^x6^  ;  W  at  (6J4,  3^).  Ans.  i6°so'  and  22°so'. 
361.— 424x724  ;  R  at  (3%,  2%).    Ans.  52°. 

;  K  at  (#,  2#).     Ans.  n°io'. 

;  R  at  (6J£,  4^).     Ans.  17*4$'. 
364.— 524x8}4  ;  K  at  (2#,  4^).     Ans.  67°3o'. 
365-— 5%x4}4  ;  R  at  (3%,  2#).    Ans.  i8°so'  and  28°4o'. 
366.— ii^x8}4;  d  at  (4^,  354).    Ans.  14°  and  16°. 

367-— 3J4MH;  K  at  (J^>  2%)-     Ans-  I4°3o''  and  J?0 
368.— 8x6^;  Vertex  at    (2^,  2^).     Ans.   38°45'  and 

67°  15'. 

369.— 8^x4^;  K  at  (7,  2*/2).    Ans.  16°  or  54°5o'. 

37o._9x5j4;  K  at  (3,  3^). 

371. — 4^x6^2  ;  Intersection  of  G.  L.  with  V  proj.  of 
axis  at  (i%,  4^)-  Ans-  2O°47'- 

372. — 9^x7;  Intersection  of  G.  L.  with  V  proj.  of  axis 
at  (7%,  4^).  Ans.  15°. 

373. — 10^x7^;  Intersection  of  G.  L.  and  V  proj.  of 
axis  at  (9^,  4%).  Ans.  17° 7'. 

374. — 524x6*4  ;  Intersection  of  G.  L.  with  V  proj.  of 
axis  at  (2^5,  4^).  Ans  n0^'. 

375. — 13^x7^  ;  Intersection  of  G.  L.  with  V  proj.  of 
axis  at  (8^,  5%).  Ans.  I2°28'  or  3°  15'. 

376. — 16x12;  Intersection  of  G.  L.  with  V  proj.  of  axis 
at  (l6#,-8#).  Ans.,  (a)  2.45"  or  io.o7/;,  (b)  11°. 

377.— 2^x4%;  V  at  (%,  2/2). 

378.— 7X6J4;  K  at  ( 

379-— 7^x6^;  &'  at 

380.— ioj4x9J^  ;  S  at  (2^,  4^).    Ans.  0.96". 

381.— 10^x6^;  S  at  (3%,  3^).     Ans.  3.79". 

382.— 4x5;  *£/  at  (2,  2^).    Ans.  59°5o'. 

383.— 4^x7^4;  ^  at  (#,  4^)-     Ans.  79°  15'. 

384.— 4^x8^;  7c^  at  (2^,  4^).     Ans.  ii°25'. 


80  PROBLEMS    IN 

385. — 73/2x10;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 
at  (4%,  6y2).  Ans.  42°59'. 

386. — 5x854  ;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis  at 
(2#,  5%).  Ans.  45°43'- 

387. — 6x9%;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis  at 
(3?4,  6#).  Ans.  46°48/. 

388.— 8x7;  e'  at  (4,  35^). 

389.— 85^x55^  ;  K  at  (2^,  3%).     Ans.  5°. 

390.— 7x754;  R  at  (3%,  4%). 

391.— 41/2x6%  ;  T  at  (i^,  3%).    Ans.  76°5o'. 

392.— 254x454  ;  d  at  (i#,  i#).     Ans.  9°55'. 

393-— 3^*75  T  at  (ift,  4).     Ans.  8303o'. 

394.— 75^x9^  ;  W  at  (154,  3^4).     Ans.  3.02". 

395.— 4^x2^;  K  at  (i%,  i%).     Ans.  3.18". 

396.— 3%x354;  R  at  (2^,  i^).     Ans.  2.01". 

397-— 4MX6%;  o'  at  (2^,  2^).    Ans.  2.43". 

398.— 4^x6%;  a'  at  (i^,  3^).     Ans.  0.55". 

399.— 654x9^;  a'  at  (i^,  5).     Ans.  1.50". 

400. — 6x654  ;  cf  at  (4%,  2^).    Ans.  1.47"  and  2.83". 

401. — 12^2x8;  a'  at  (254,  5).     Ans.  io°4O/. 

402.— 6^x7;  a'  at  (2^,  3%).     Ans.  23°  15'. 

403.— 6x7;  a'  at  (354  3%).     Ans.  2i°3o'. 

404. — 8xio ;  a1  at  (4}^,  3). 

405-— 9>4X75  «•'  at   (2%,  354).     Ans.  7°2or  and  67°3Or. 

406.— 135^x10;  c'  at  (4,  454).     Ans.  ii°i5'  and  I3°5o' 

407.— 1154x65^;  R  at  (8%,  35^).    Ans.  4*45'  and  io°is'. 

408.— 6x5^4. ;  ar  at  (25^,  3^).    Ans.  I5°io'  and  io60i5r. 

409.— 13)4x5;  G  at  (i%,  3%).    Ans.  i3°45'  and  36°i5r. 

410.— 1134x115^  ;  V  at  (45^,  8).     Ans.  2°so'  and  n°i5'. 

411.— 754x7^;  a'  at  (2%,  4).     Ans.  7o°s'. 

412.— 954x954  ;  a'  at  (254,  3%).     Ans.  53°i5'- 

413. — 16x12;  b'  at  (7J^,  6). 

414.— 15x12;  V  at  (45^,  6). 

415.— 16x12;  S  at  (734,  654). 

4I6._i6xi2;  V  at  (754,  854). 

417.— 4#X7J£;  a'  at  (2# 


DESCRIPTIVE   GEOMETRY  8  1 


418.—  5^x5;  »'  at   (i%, 

419.—  3tt*4lA  ;  o  at  (*/&>  2 
420.—  4/2x3^  ;  R  at  (33/4, 

421.—  3^x434;  K  at  (-H, 
422.—  5x4%;  K  at  (3/4,  2i 
423.—  2^x4  y4  ;  K  at  (M, 
424.—  43/4x81/4  ;  K  at  (13^, 
425.—  6j4x;>4  ;  <?'  at  (3^, 
426.—  3^x534;  o  at  (23/6,  i5/&). 

427.  —  15x1  1*/;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 
at  (2%,  534).     Put  lower  left  hand  corner  of  development 
at  (2y&,  8J4). 

428.  —  15^2x6^2;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 
at  (2-Hj,  334).     Put  lower  left  hand  corner  of  pattern  at 


429.  —  10x9^  ;   Put  construction  work  on  left,  and  pat- 
tern on  right. 

430.  —  nj4xn%;  R  at  (2,  6%). 

431.  —  11x15;  S  at  (2%,  6^4).   Middle  of  developed  right 
section  at    (4J/2,    13).     Ans.   Length  of  pattern  =  8.69". 
Length  of  longest  element  =  2.37". 

432.  —  iixio  ;  K  at  (i^4,  6j^).   Middle  of  developed  right 
section  at  (10%,  5^2).     Ans.  Length  of  pattern  =  8.64". 

433-—  1524x8%;  Center  of  base  at  (ij4,  ij^). 
434-—  7l/2Xi4y2  ;  Vertex  at  (5^,  9^).     Ans.   (b)   maj. 
=  3.48",  min.  ==  2.86",  (c)  161°. 

435.  —  4j4x6j^  ;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 
at  (2}4,  2#). 

436.  —  12x7^4  ;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 
at  (2%,  SJ^).     In  pattern,  put  vertex  at  (8%,  7). 

437.  —  io*4x624  ;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 
at  (i%,  3^)-     In  development,  put  vertex  at  (6%,  6^). 

438.  —  14^/2x1024  J  R  at   (J^,  4^4).     Vertex  in  develop- 
ment at  (ioj4,  10^). 

439.—  13x8^;  R  at  ( 


82  PROBLEMS    IN 

440.—  14^x8%;  T  at  (ys,  4^).     Vertex  for  pattern  at 


441.—  13x7^4;  R  at  (5%, 

442.—  &y2xi3y2  ;  »'  at  (3%, 

443.  —  16x11  ;  Center  of  developed  helix  at  (4%,  8). 

444.—  4>4x8^;  ri  at  (i^,  5^). 

445-  —  6^x6^;  Center  of  developed  helix  at  (4,  4). 

446.—  7^x6^4  ;  «'  at  ( 

447-—  6^x7%;  a'  at  ( 

448.—  8j4xii;  a'  at  (2^,  3%). 

449-—  6^x5;  Rat  (#,  2#). 

450.—  6^4x6^;  R  at  (#,3). 

451.—  10x12;  R  at  (4%, 

452;—8Mx6^;  nr  at  (2 

453.  —  Let  the  floor  be  taken  coincident  with  the  H  plane 
of  proj. 

454.—  ioj^xi6;  n'  at  (6%,  6>^). 

455.  —  11x12;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis  of 
cone  at  (6^4,  4%).  Ans.  Z  at  vertex  of  developed  cone 


456.  —  14x10^4  \  Intersec.  of  tank  axis  and  G.  L.  at  (2, 
:  Lower  left  corner  of  developed  tank  at  (4^5,  o)  ; 
Lower  left  corner  of  developed  pipe  at  (6,  5)  ;  Vertex  of 
developed  cone  at  (14^,  7^)-  Ans.  Length  tank  pattern 
=  12.57";  Length  pipe  pattern  =  4-7i";  Angle  at  vertex 
of  developed  cone  =  254°25'. 

457-—  13^x9;  K  at  (6y2, 

458.—  10x16;  V  at  (4^, 

459.  —  15x10^4;  Lower  left  hand  corner  of  Fig.   17  at 


460.—  13^x10^  ;  Put  G.  L.  4"  tip. 

461.  —  io^4xioj4  ;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 

at  (5%,  5#)- 

462.—  7^4x15;  d'  at  (s#,  8J4). 

463.  —  9T4xn;  Intersec.  bet.  G.  L.  and  V  proj.  of  axis 
of  ist  cone  at  (4^4, 


DESCRIPTIVE   GEOMETRY  83 

464. — 11x16;  c'  at  (5*4,  5*4).  Middle  of  developed 
cylindrical  base  at  (n,  IT).  Vertex  of  developed  cone  at 
(o,  16). 

465. — ioj4x9J/2.  Intersec.  bet.  G.  L.  and  V  proj.  of 
axis  at  (8%,  4).  Vertex  of  developed  cone  at  (r/s,  4-)4)- 
Ans.  /  at  vertex  of  developed  cone  ==  109° 9'. 

466. — 14x10;  n'  at  (6*4,  7). 

467.— 6x6;  <?'  at  (3^, 

468.— 8x954;  Kat  (2j4, 

469.— 14x8^4  ;  K  at  (2J4, 

470.— 5J4x8j4;  a'  at  (2^,  3%). 

471.— 5^x7;  a'  at  (2%,  3%). 

472.— 5x554;  a'  at  (i^,  3Ji). 

473-— 6^/4x9^;  H  proj.  of  axis  of  torus 'at  (3^, 

474.— 4*4x7%  ;  ar  at  (19^, 

475-— 3J4*5^  :  n/  at 
476.— 5/2x6^4  ;  &'  at 
477.— -i 1 14x4^4  ;  6'  at  (414, 
478.— 5^x8>4;  S  at  (1*4, 
481. — 5^4x11^2;  Intersec.  bet.  G.  L.  and  Y  proj.  of  axis 
of  helix  at  (2^,  514). 
482.— 4^x554  :  &'  at  (%,  254). 


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